MSU LIBRARIES asl- RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped be1ow. The Kinetic Effects of the Binding of Mitochondrial Creatine Kinase to Chicken Heart Inner Mitochondrial Membranes By Stephen Philip Joseph Brooks A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Biochemistry 1986 ABSTRACT The present study examines the relative contributions of the outer mitochondrial membrane, and the binding of mitochondrial creatine kinase (MiMi-CK) to the inner mitochondrial membrane (IMM), to the preferential coupling of MiMi-CK to oxidative phosphorylation in chicken heart mitochondria. Mitochondrial creatine kinase (EC 2.7.3.2) was purified to homogeneity from chicken ventricle using a procedure which makes use of an Agarose-Hexane-ADP column run under conditions where a MiMi-CK active site transition-state analog is formed. The homogeneous enzyme has a Mr = 86,000 1 5,000 and a specific activity of 12“ IU/mg. An examination of the binding process shows that the MiMi- CK:IMM interaction is ionic in nature. The binding process is dependent on the pH of the incubation medium with increased binding at lower pH values; binding is dependent on the protonation of a group(s) with an apparent pKa value of 6. Extrapolating titrations of mitoplasts with MiMi-CK gives a maximum of 14.6 IU bound per nmole cytochrome 33 on the IMM. This value corresponds to 1.12 moles MiMi- 3 CK per mole cytochrome as or 0.33 mole MiMi-CK per mole inner membrane 3 nucleotide translocase. The kinetic parameters of MiMi-CK were examined using either intact mitochondria or mitoplasts. When MiMi-CK is coupled to the 2 of 36 H! is obtained. nucleotide translocase. a Ka value for MgATP- This Ka value is three fold lower than that measured using mitoplasts under conditions where about 70% of the enzyme is bound to the IMM (100 ll!), or for that of soluble enzyme (125 ufl). The apparent nucleotide translocase Km value for ADP decreases from 20 ufl to 10 uM in the presence of 50 mM creatine only when intact mitochondria are used. Using these different kinetic constants and coupling enzyme theory, one can correctly predict the steady state concentrations of ATP and ADP indicating that the measured Km values reflect the solution kinetic parameters of the system. These results show that preferential coupling is dependent on the presence of the outer mitochondrial membrane in chicken heart mitochondria. To my wife Janet for her patience and her love. 11' ACKNOWLEDGEMENTS I would like to thank all those who have contributed either directly or indirectly to this dissertation. Although it is impossible to name everyone, I would like to mention the members of the lab (past and present) Peter Toth, Vickie Bennett-Hershey, Jeff Baxter, and James Dombrowski for help and ideas. I would especially like to thank the members of my guidance committee, Dr. R. Anderson, Dr. G. Babcock, Dr. S. Ferguson-Miller, and Dr. J. Wilson for their patience, comments and support. As for thanking Dr. Clarence Suelter - I simply cannot say enough. 1'1'1' TABLE OF CONTENTS Page List of Tables vii List of Figures viii Abbreviations xi Introduction 1 Chapter I: Literature Review 3 Kinetic Studies of Creatine Kinase 5 Cellular Localization of Creatine Kinase 8 The Creatine Phosphate Shuttle 10 MiMi-CK and Oxidative Phosphorylation 1N MM-CK and the Myosin ATPase 20 References Cited 20 Chapter II: Theory and Practical Application of Coupled 29 Enzyme Reactions: One and Two Auxiliary Enzymes Abstract 29 Introduction 29 Theory 31 One Auxiliary Enzyme 31 Two Auxiliary Enzymes 38 Minimum Values of V2 and V3 A6 Discussion 50 Acknowledgement 52 References 52 Appendix 53 One Auxiliary Enzyme 53 Two Auxiliary Enzymes 58 Program Minimum 59 Chapter III: Theory and Practical Application of Coupled 62 Enzyme Reactions: One and Two Coupling Enzymes with Mutarotation of an Intermediate Abstract 62 Introduction 63 Materials and Methods 6A Theory 65 One Auxiliary Enzyme 65 Two Auxiliary Enzymes 70 Minimum Values of V2 and V3 7" Results 75 iv Discussion 82 References 82 Appendix 85 One Auxiliary Enzyme 85 Two Auxiliary Enzymes 89 Chapter IV: Characterization of Chicken Heart Mitochondrial 91 Creatine Kinase Using Transition State Analog Affinity Chromatography. Abstract 91 Introduction 91 Materials and Methods 93 Materials 93 Enzyme Assays and Protein Determinations 9A Cellulose Acetate Electrophoresis 9M Polyacrylamide Gel Electrophoresis 9U Sequencing and Amino Acid Analysis 95 Equilibrium Centrifugation 95 Carboxypeptidase Y Digestions 96 Determining Kinetic Constants 96 Purification Procedure 96 Isolating Mitoplasts 97 Releasing MiMi-CK from Mitoplasts 97 Procion Red-agarose Chromatography 97 Affinity ChromatOgraphy 98 Concentrating and Storing the Enzyme 10A Results 10A Procion Red-Agarose Chromatography 1ou Transition State Analog Affinity Chromatography 105 Purity and Extinction Coefficient 105 Amino Acid Composition and Sequences 110 C-Terminal Studies 111 Kinetic Studies 111 Molecular Weight Studies 115 Discussion 115 References 118 Ctuapter V: Association of Avian Mitochondrial Creatine 122 Kinase with the Inner Mitochondrial Membrane Abstract 122 Introduction 123 Materials and Methods 125 Preparation of Avian Heart Mitoplasts 125 Purification of MiMi-CK 126 Titrating Mitoplasts with MiMi-CK 126 Titrating Adenine Nucleotide Translocase 126 with Carboxyatractyloside Titrating Mitoplasts with Adriamycin 127 Other Procedures 127 Results 128 Effects of Neutral Salts on the Interaction 128 of MiMi-CK with Mitochondria and Mitoplast Preparations Effect of Substrates on the Interaction of 131 MiMi-CK with Mitoplasts Stoichiometry and Dissociation Constant for 135 the Interaction of MiMi-CK with Mitoplasts Effect of Adriamycin 1A5 Effect of Osmotic Strength 150 Discussion 150 Acknowledgments 153 References 15A Chapter VI: Preferential Coupling of Mitochondrial Creatine 158 Kinase to Chicken Heart Nucleotide Translocase in Mitochondria and Mitoplasts Abstract 158 Introduction 159 Materials and Methods 160 General 160 Preparation of Mitochondria and Mitoplasts 161 Enzyme Assays 161 Measuring Creatine Phosphate and ATP 163 Concentrations Theory 163 Results 167 Binding of MiMi-CK to Mitoolasts 167 Kinetic Constants of MiMi-CK 170 Kinetic Constants for the Nucleotide Translocase 171 Rates of Creatine Phosphate Synthesis 175 Measuring ADP Concentrations 178 Discussion 181 References 18A Chapter VII: General Conclusion 186 Appendicies Appendix A: Listing of WILMANM Appendix B: Listing of LAGTIME Appendix C: Listing of NONLIN Appendix D: Published and Submitted Papers vi Table Table Table Table Table Table Table Table Table Table Table Table II-1 IV-1 IV-2 IV-3 IV-H IV'S V-1 VI-1 VI-2 VI-3 LIST OF TABLES Comparison of t values calculated by various methods For the two auxiliary enzyme system. Purification summary for the preparation of MiMi-CK. Measurements of MiMi-CK protein by spectrophotometric methods. Amino acid composition of MiMi-CK. Kinetic constants for MiMi-CK. Characteristics of MiMi-CK purified from various sources. and C values for the release of MiMi- CEO from mi§0plasts by various salts at pH 7. A. Specific activity of MiMi-CK released by various salts at 2 X 050 concentrations. and C values for the release of MiMi- CEO from mi§oplasts by enzyme substrates. Kinetic constants in m! for MiMi-CK. Determination of the MiMi-CK Km value for ATP and the ATP steady state concentration. Steady state ADP concentrations for different mitochondrial preparations. vii Page AU 99 106 107 11“ 117 13A 136 137 172 176 177 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure II-1 II“2 II-3 II-N II-5 II-6 II-Ala II-A1b III-1 III-2 III-3 III-Ha III-Nb III‘S LIST OF FIGURES Relationship between product concentration and time for a theoretical enzyme assay. Theoretical prediction from Equation II-7. Theoretical relationship between log(v /V2) and the time required for B to reach 99% of its steady state value. Relationship between t and log(v1/V ). 99 3 Graphical representation of the cost minimization technique. Relationship between t and T. FC Graph of [BJ/K versus reaction time calculated by Ehree methods. Plot of t versus F calculated from various equations. Theoretical time course for a one enzyme coupled assay involving mutarotation of the intermediate as shown in Scheme 1. Graph of V calculated from Equation III-23 versus the volume of glucose 6-phosphate dehydrogenase added to the reaction mixture. Comparison of theoretical and actual results for the hexokinase-glucose 6-phosphate dehydrogenase (G6 PdH) system. Titration of glucose 6-phosphate dehydrogenase with phosphoglucomutase. Experimental progress curves for measurement of phosphoglucomutase enzyme activity. /V . Relationship between t99 and v1/V2 or v1 3 viii Page 12 33 33 HO A0 A8 57 57 66 77 77 81 81 81 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure III-Ala III-A1b IV-1 IV-2 IV’3 IV-H IV-5 V’3 V-H V-S V-6a V-6b V'7 V-8 Theoretical time courses for the accumulation of [8] calculated from three sources. Relationship between the error obtained from the t analysis and the true tFB value as a functIgn of v1/V2. ADP (Transition state analog) column profile. SDS-Polyacrylamide gel electrOphoresis of various stages of purification. N-Terminal sequences for creatine kinase from chicken tissue. Kinetic activity and binding ability of carboxypeptidase I treated MiMi-CK. Equilibrium sedimentation determination of MiMi-CK molecular weight. Release of MiMi-CK from mitochondria and mitoplasts. Effect of increasing salt concentrations on the percentage of MiMi-CK released from mitoplasts. Release of MiMi-CK from mitoplasts with increasing substrate concentrations. Association of MiMi-CK with mitochondrial membranes. Binding of purified MiMi-CK to mitoplasts: Klotz plot. Effect of pH on the apparent K value for MiMi-CK binding to mitoplasts. Plot of the log of the Kapp value against the reciprocal of the hy ogen ion concentration used to determine the values of K2 and K3. Effect of adriamycin on the binding of purified MiMi-CK to mitoplasts. Binding of adriamycin to MiMi-CK. Swelling of mitoplasts at reduced ix 88 88 101 103 109 113 113 130 133 133 139 139 1N3 1M3 1116 1H6 1N9 Figure VI-1 Figure VI-2 Figure V-3 Figure VI-A Figure VI-S osmolalities. Coupling the synthesis of creatine phosphate to oxidative phosphorylation through MiMi- CK. Binding of MiMi-CK to the inner mitochondrial membrane. Measuring the kinetic parameters for the synthesis of creatine phosphate by MiMi-CK. Effect of increasing creatine concentrations on the V and K for the nucleotide m x m translocase. Rates of creatine ph03phate synthesis with mitochondria and mitoplasts. 166 169 17A 180 180 ABBREVIATIONS ApSA: P',P -Di(adenosine-5')pentaphosphate 5 BB-CK: brain type creatine kinase BICINE: N,N-bis(2~hydroxyethyl)glycine BSA: bovine serum albumin CK: creatine kinase DTT: dithiothreitol EGTA: ethyleneglycol-bis-(2-aminoethyl ether) N,N,N',N'-tetraacetic acid G6PdH: glucose 6-phosphate dehydrogenase Hepes: N-2-Hydroxyethylpiperazine-N'-2-ethanesulfonic acid IMM: inner mitochondrial membrane IU: 1 umole of substrate converted per minute MES: 2-(N-morpholino)ethanesulfonic acid MiMi-CK: mitochondrial creatine kinase MM-CK: muscle type creatine kinase MOPS: 3-(N-morpholino)propanesulfonic acid NaDOC: sodium deoxycholate OMM: outer mitochondrial membrane PGM: phosphoglucomutase PMSF: phenylmethylsulfonyl fluoride TPCK: N-tosyl-phenyalanine chloromethyl ketone xi INTRODUCTION The localization of the mitochondrial isozyme of creatine kinase (MiMi-CK) in between the inner and outer mitochondrial membranes reprotedly provides the enzyme with preferential access to the ATP generated by oxidative phosphorylation. This preferential access is demonstrated by lower MiMi-CK kinetic constants for MgATP-z when the enzyme is coupled to oxidative phosphorylation as compared with the kinetic constants of the soluble enzyme by direct assay of the products. Preferential access is also demonstrated by trapping experiments where ATP generated by oxidative phosphorylation is utilized preferentially over the solution ATP. Two prevailing hypotheses can account for the observed preferential access. A. Binding of MiMi-CK to the nucleotide translocase aligns the active sites of the two enzymes such that MiMi- CK sees a locally high ATP concentration. B. The outer mitochondrial membrane limits the diffusion of the nucleotides so that the localization of MiMi-CK in the inter membrane space exposes the enzyme to a higher ATP concentration than is found in the cytosol. In order to distinguish between these two hypotheses, chicken heart mitochondria which contain a low amount of MiMi-CK were examined for the relative contribution of the binding of MiMi-CK to the inner mitochondrial membrane (IMM), and the outer mitochondrial membrane to functional coupling. Chicken heart mitochondria are used so that the rate limiting step in the coupled MiMi-CK:nucleotide translocase reaction is MiMi-CK. In order to measure the kinetic constants for bound MiMi-CK the conditions under which MiMi-CK is bound to the IMM had to be defined. Examining the binding process required the determination of the dissociation constant for the enzyme, a value which required the titration of IMM with homogeneous MiMi-CK. This dissertation presents the results of an examination of the preferential coupling of MiMi-CK to oxidative phosphorylation in chicken heart mitochondria. The dissertation is organized into seven separate chapters. The first chapter is a review of the pertinent literature. Chapters II to VI are written as separate papers, each with their own Introduction, Materials and Methods, Results, Discussioo and References sections. The last chapter is a general discussion and summary. Chapters II and III deal with coupled enzyme theory and are presented to give a background to the results presented in Chapter VI. Chapters IV and V contain the results of the purification and binding studies of MiMi-CK. Chapter VI presents the results of the experiments on the preferential coupling of MiMi-CK to oxidative phosphorylation. Chapter I Literature Review Eppenberger 22 El. (1, 2) first discovered the existence of two cytosolic creatine kinase (EC 2.7.3.2) isozymic variants. These isozymes, M (muscle type) and B (brain type) are located in various tissues in different species. The M isozyme is found in mature mammalian and avian skeletal muscle and mammalian myocardium, the B isozyme in mammalian brain, neural tissue, and embryonic skeletal muscle and avian myocardium (1 - 3). A third isozymic form, Mi (mitochondrial type), is found in large amounts in human, beef, and rat heart, as well as rat brain, skeletal muscle, and intestinal muscle mitochondria (A, 5). The largest amount of mitochondrial creatine kinase per mitochondrion is found in chicken breast muscle, a pure white fiber muscle (6). Initially the Mi isozyme was not found in chicken heart tissue (7) but subsequent studies have confirmed that the isozyme is present at about 1/20 of the chicken breast muscle concentration (6). The assignment of creatine kinase as B or M type is based on the initial localization of these enzymes in brain and skeletal muscle of rabbit (8). The differentiation of these isozymes is based on their migration in starch gel electrophoresis (7. 9), cellulose acetate electrophoresis at pH 8.8 (6, 10) or isoelectric focusing (11). At pH 8.8, the B type migrates quickly to the positive terminal, the M type is neutral or slightly positive, and the Mi type moves toward the negative terminal. This general pattern is seen for all species although the 3 extent of migration may differ. Studies from Kaplan's laboratory (1, 2, 9) demonstrated that intact creatine kinase is a dimer. They showed this by renaturing mixtures of M and B type enzymes from various tissue sources and animal species. The resulting starch gel electrophoretic patterns showed three protein bands which are assigned as the original dimeric MM and BB enzymes and a mixed type, MB, with a pI value exactly in between the MM and BB isozymes. Analyzing several of these creatine kinase mixtures shows that the subunits mix in equal (1:1) stoichiometries. Determining the molecular weight of the intact enzyme by exclusion chromatography, and of the subunits by SDS-polyacrylamide electrophoresis (12 - 16), confirms the dimeric structure of creatine kinase. Because of the above results, muscle type creatine kinase is abbreviated as MM-CK, brain type BB-CK and mitochondrial type MiMi-CK. These abbreviations will be used throughout the rest of this dissertation. Interestingly, a unique isozyme of creatine kinase which is a monomer with a Mr of 1N7,000 is located in the tail of sea urchin sperm (17). Although the intact enzyme is a dimer, isolated subunits apparently have a specific activity equal to that of the intact enzyme (18) suggesting that a dimeric structure is not necessary for catalytic activity (19 - 21). In the dimeric enzyme the subunits show negative COOperativity with respect to MgADP- binding (22) indicating that conformational changes in one subunit are transmitted to the other. Individual cytosolic subunits readily Join with one another both 13,3129 (7) and $2.!l322 (1, 2) to form the MB isozyme but the mitochondrial subunit does not hybridize with either of the two cytosolic forms (13, 1h, 23 - 25). Amino acid sequence data and studies of 5 antibody cross-reactivity show that the cytosolic subunits are similar in their N and C terminals (26 - 28), and that antibodies directed against one cytosolic subunit react with the other (1”, 29). The mitochondrial isozyme, on the other hand, is clearly different: antibodies against the cytosolic isozymes do not react with the mitochondrial isozyme (10, 13, 1M, 29) and N-terminal sequence analysis shows a large difference between cytosolic and mitochondrial sequences (27. 28). The molecular weight of the M subunits appears to be identical at Mr - “3,000 for the human, canine, rabbit mouse and bovine enzymes but the B subunit molecular weights are different: human, AN,500, canine, N6,000, rabbit, AA,OOO, and mouse, u9,ooo (29). The molecular weight values for dimeric MiMi-CK are reported as follows: beef heart, 65,000 (15), human heart, 82,000 (1M) and 8A,OOO (25), and dog heart, 8H,OOO (23). Kinetic Studies of Creatine Kinase Creatine kinase catalyzes the reversible transfer of the phosphoryl group from ATP to creatine as shown in equation I-1 (30, 31). [1-1] MgATP_2 + creatine < — — + > MgADP + creatine phosphate 2 + H The kinetic properties of the enzyme were established using purified skeletal muscle (M isozyme, 1, 32) and brain (B isozyme, 2) preparations. The pH optimum for the forward reaction (creatine phosphate synthesis) is approximately 8 - 9 and for the reverse direction (creatine synthesis) is about 7 (33. 3A). The effect of divalent cations on the reaction kinetics showed that MgATP-Z and MgADP- are the substrates for creatine kinase (33. 35. 36) but other divalent cations such as Mn+2 (the rate is 6 about 75% of the rate in the presence of Mg+2) and Ca+2 (the rate is about 25% of the rate in the presence of Mg+2) can effectively substitute for Mg+2 in both the forward (37) and reverse (38) directions. Other nucleoside diphosphates are capable of binding the enzyme but all (except deoxy ADP) have a five to ten fold lower affinity and Vmax value (39). The kinetic mechanism was determined by using classical kinetic studies. Product inhibition data indicate that for the forward reaction MgADP_ is competitive with MgATP-2 and nonrcompetitive with creatine, and creatine phosphate is competitive with creatine and non-competitive with MgATP-2 (33, 38). A similar pattern, seen for the reverse reaction, combined with the postulated existence of the two dead end complexes shown in equation I-2 (38), and substrate binding data (8, 36) indicate that the reaction obeys a rapid equilibriun random mechanism. Later studies have confirmed this for the mitochondrial isozyme as well (A0). [I-Z] -2 /—> CK:MgATP :CrP <—-\ -2 CK: MgATP CK:CrP MgATP z/Ka Kb\ Cr MgADP- Kd ROCK\ CrP v CK CK: MgATP2 :Cr <—§> CK:MgADP :CrP v ’ ’2 l" r \Kb Ka MgATP CrP \Kc K/dCK MgADP CK:Cr CK:MgADP \———> CK:MgADP-:Cr <.—/ In equation I-2 Cr and CrP represent creatine and creatine phosphate, 7 respectively. The constants Ka, Kb’ Kc’ and Kd 2, Cr, CrP and MgADP- binding to the enzyme and the are the dissociation constants for MgA'I‘P- constants Ka, K K and K represent the dissociation constants in the b’ c d presence of the other substrate. Although several of the initial kinetic studies were carried out in chloride containing buffers, later observations of the effects of anions such as 01-, HP0;2, and SD;2 demonstrated that these ions are competitive inhibitors of creatine phosphate and non-competitive inhibitors of MgADP- (35, 36). A further examination of this phenomenon showed that the chloride ion sits in between the substrates in the CK:MgADP-:Cr dead end complex to form a very stable quaternary complex (37). Planar anions such as N03- and N02- are the most effective in stabilizing the complex: the K value for MgADP- binding to the dead end D complex is seven fold lower in the presence of Cl— and thirty fold lower 3 O (37) to prepose that planar anions mimic a planar phosphoryl group and in the presence of NO The above results led Milner-White and Watts thus MgADP-:anion:Cr forms a transition state analog. This explanation accounts for the large decrease in the K values for MgADP- in the D 3f Studies of the active site show the presence of a cysteine sulfur presence of Cl- and NO group which must be reduced for maximal enzyme activity (38, A1) but which apparently does not participate directly in the reaction sequence (A2). A histidyl residue (A3, AA) and a lysyl residue (A5) which are apparently involved in the reaction sequence have been reported. The reaction occurs by a direct in line transfer of the phosphoryl group from one substrate to the other in the active site (A6, A7); a phosphorylated enzyme intermediate could not be detected (33. 3A). Cellular localization of creatine kinase Many "soluble" enzymes bind reversibly to cellular structures (A8). These enzymes, termed ambiquitous to describe their dual bound/free status (A9), frequently exhibit altered kinetics when in the bound versus free state (A8). Both M type and M1 type creatine kinase isozymes are ambiquitous. Muscle type creatine kinase apparently binds to the M-line of skeletal muscle tissue as demonstrated by the localization of antibodies directed against MM-CK by immunofluorescence (50) and by the release of creatine kinase activity from muscle fibers after treatment with antibodies directed against the M line proteins (51). lfl.l££§9 studies of MM-CK binding to myosin and to myosin subfragments generated by protease action, confirm that MM-CK binds to the tail portion of myosin (52, 53) but the number of MM-CK binding sites was not measured. Washing myofibrils with low ionic strength buffer shows that 5% to 10% of the total cytoplasmic creatine kinase activity is bound to the M-line of myofibrils in chicken pectoralis (5A, 55). Brain type creatine kinase apparently does not bind to chicken heart myofibrils (5A, 55). Examining the effect of increasing salt concentrations on heart mitochondria and outer membrane stripped mitochondira (mitoplasts) led Scholte gt él- (56) and Jacobus and Lehninger (5) to localize MiMi-CK on the outside of the inner mitochondrial membrane (IMM). These results were later confirmed by Iyengar and Iyengar (57) using sonicated mitochondria and detergents to localize the enzyme. Despite these findings, studies of the binding and release of MiMi-CK are still performed with intact mitochondria (6, 58 - 61). They show that MiMi-CK 9 is released from the mitochondria by increasing ionic strength. Presumably, the release of the enzyme from mitochondria is due to a ruptured outer mitochondrial membrane which permits the passage of MiMi- CK through large holes and consequently the data on the release of MiMi- CK from the IMM are not easily interpretable. Furthermore, the above studies (58 - 61) fail to provide information regarding optimal binding conditions, the dissociation constant for MiMi-CK, or the number of MiMi- CK binding sites on the IMM. A study by Hall and DeLuca (62) examined the binding of semi-purified MiMi-CK to phosphate extracted mitochondria. Although they observed saturable binding, they failed to quantitate the data, nor did they demonstrate that the enzyme associated exclusively with the IMM. Interestingly, sulfhydryl reagents such as paga- chloromercuribenzoate release the enzyme from mitochondria (59. 60) but whether sulfhydryl residues participate directly in the binding process has not been demonstrated (61). Recent attempts to define the binding site for MiMi-CK on the IMM led Muller gt 2l° (6A) to conclude that MiMi-CK binds to cardiolipin in the IMM. Their studies showed that MiMi-CK bound to cardiolipin vesicles but not to phosphatidylcholine vesicles. They also demonstrated that MiMi-CK is released from mitoplasts by adriamycin (63, 6A), a cardiolipin-binding drug. Their data was later supported by Schlame and Augustin (65) who treated mitochondria with phospholipase A2 and phospholipase C and compared the amount of MiMi-CK which was solubilized. Since only phospholipase A released MiMi-CK, they concluded that 2 cardiolipin was the MiMi-CK binding site. The above results are strengthened by the observation that MiMi-CK binds liver mitoplasts (which contain no MiMi-CK). This latter observation suggests that the 10 binding factor for MiMi-CK is present in mitochondria which do not have endogeneous MiMi-CK (62, 6A). However, not all the data indicates that the MiMi-CK receptor in the IMM is cardiolipin. Experiments by Vial gt El° (58) show that only 6% of the total MiMi-CK binds to sonicated mitoplasts (inverted vesicles) under conditions where 51% of the total MiMi-CK binds to mitoplasts. If MiMi-CK binds only to cardiolipin on the IMM, then the inside of the IMM should bind at least the same amount of MiMi-CK as the cytosolic side of the IMM; the matrix side has three times more cardiolipin than the cytosolic side (67). Furthermore, Kuznetsov and Saks (66), who report a MiMi-CKztranslocase ratio of around 1:1, suggest that MiMi-CK binds directly to the translocase, and not to cardiolipin (67). The Creatine Phosphate Shuttle The importance of creatine, and the creatine kinase reaction (equation I-1) to muscle bioenergetics is demonstrated by the high concentration of creatine phosphate found in muscle tissue (68, 69) and the direct correlation between creatine phosphate breakdown and muscle contraction (30, 70). Although initial investigators believed that creatine phosphate was the direct energy source for muscle contraction, the discovery of ATP (71, 72), and the observation that ATP stimulated actomyosin contraction (73 ’ 75) showed that adenine nucleotides were involved in the contractile process. These results, coupled with the measurement of ATP hydrolysis in contracting muscle tissue poisoned with 2,A-dinitrofluorobenzene (which inhibits creatine kinase activity) demonstrated that ATP directly supplies the energy for contraction (76). In the present model of muscle contraction, ATP hydrolysis 11 Figure I-1: Schematic representation of the creatine phosphate shuttle. 12 \ r .E/ \6 i 1} means: 5.5: J . / L I ao<\ mo 5.68523 k III n.._.< .2/11\. 6.22 IXE\ /%4 23325 13 indirectly provides the source of energy for contraction; the largest free-energy change results from inorganic phosphate release from the S-1 protein (actin dependent myosin ATPase) after ATP hydrolysis (77. 78). Regeneration of ATP at the myosin ATPase can be accomplished by adding external ATP (73 - 75) or creatine phosphate (79) as shown by the initial experiments on muscle contraction, 69 - 80). The first evidence for the creatine phosphate shuttle came from experiments which show that ATP and creatine phosphate are compartmentalized and not in equilibrium as suggested by the above experiments. Studies by Gudbdarnson gt al. (82) and Seraydarian 33 al. (83, 8A) demonstrate that the ability of ischemic heart muscle or cultured heart muscle cells to contract correlates with the concentration of creatine phosphate and not ATP. Gudbjarnson 93 al. (82) assayed creatine phosphate and ATP concentrations at different time points after the onset of ischemia. At the one minute time point, the heart stopped beating even though over 80% of the original ATP concentration remained. Interestingly, at one minute, over 70% of the creatine phosphate had been hydrolyzed. This indicated to the authors that a "functional compartmentation of ATP and creatine phosphate" existed in dog heart muscle since it appears that not all of the ATP has access to the myosin ATPase. These conclusions are supported by Seraydarian (85) using frog sartorious muscle. She showed that fatigued muscle (which does not respond to electrical stimulus) still has more than 80% of its original ATP but less than 30% of its creatine phosphate. The proposal for compartmentation of creatine phosphate was strengthened by pulse-chase experiments with [1-1"C] creatine which suggested that two separate pools of creatine phosphate exist in heart tissue (see 86). 1A The above data led Bessman to propose the existence of a creatine phosphate shuttle in muscle tissue (86, 87). The central idea of this shuttle mechanism is that creatine phosphate serves as a high energy phosphate carrier which shuttles between the site of ATP synthesis (mitochondria) and ATP breakdown (myosin ATPase). Thus the mitochondrial creatine kinase and muscle creatine kinase, in mammalian muscle, serve different functions within the cell (Figure I-1). When muscle contracts, MM-CK rephosphorylates the resulting ADP using creatine phosphate as a substrate. The creatine produced by MM-CK diffuses to the mitochondria where MiMi-CK uses ATP to rephosphorylate creatine. The MiMi-CK isozyme also provides ADP to the respiratory system stimulating oxidative phosphorylation. Support for the creatine phosphate shuttle comes from the localization of M type and Mi type creatine kinase near the energy consuming and energy producing centers in the cell, and from kinetic studies of the interaction between the creatine kinase reaction and muscle contraction or oxidative phosphorylation. MiMi-CK and Oxidative Phosphorylation The effect of the localization of MiMi-CK in the inter membrane space of mitochondria was studied initially by Jacobus and Lehninger (5). They demonstrated that MiMi-CK could accept the ATP released from oxidative phosphorylation and provide the nucleotide translocase with ADP so that creatine increases the post ADP-stimulated respiration rate (state A). Although they observed that creatine also increases the ADP- stimulated respiration rate (state 3) at lower ADP concentrations, thus showing that MiMi-CK generates a locally high ADP concentration near the nucleotide translocase, they concluded simply that the role of MiMi-CK in 15 the heart tissue is to maintain a high steady state ADP concentration near the translocase. A direct demonstration of the kinetic effect of MiMi-CK localization in the inter membrane space is provided by Saks _e_t il. (88) who showed that creatine phosphate synthesis is greater in the presence of oxidative phosphorylation that in its absence (when oligomycin A is added) even though the total ATP concentration is equal in both cases. This result suggests that a locally high ATP concentration exists near the MiMi-CK active site, as was shown for the nucleotide translocase (5). Measuring the MiMi-CK Km for MgATP-2 (in the presence of a finite creatine concentration) confirms the above conclusion: the Km value is 37 pg in the presence of oxidative phosphorylation and 200 pg in its absence even when the external ATP is regenerated by adding phospho(enol)pyruvate and pyruvate kinase to the suspension (89 - 91). This latter Km value is similar to the Km value of 1A5 ufl determined for semi-purified MiMi-CK (89). These results suggested to Jacobus and Saks that MiMi-CK prefers the ATP generated by oxidative phosphorylation over that added externally to the suspension (90). In order to test the kinetic results obtained in Saks' and Jacobus' laboratories, Erickson-Viitanen 32 El: (92, 93) measured the contribution of mitochondrial versus cytosolic ATP to the formation of creatine phosphate. Mitochondrial ATP was labelled by starting the reactions with H332POA so that the creatine phosphate formed from oxidative phosphorylation would be radioactive. The specific activity of creatine phosphate generated by MiMi-CK in the presence of increasing concentrations of unlabelled ATP, compared with the specific activity of the solution ATP, shows that at external ATP concentrations of greater 16 that 0.1 mM, less than 10% of the creatine phosphate produced comes from ATP generated by oxidative phosphorylation. However, below this value, the percentage of creatine phosphate derived from radioactive ATP (mitochondrial in origin) rises sharply to about 50% (92). The above results were challenged by Altschuld and Brierley (9A) and Borrebaek (95) who failed to observe differences in the Km values measured with oxidative phosphorylation or with a pyruvate kinase- phospho(enol)pyruvate regenerating system. However the data of Erickson- Viitanen gt a}, (92) can explain this apparent discrepancy. Both Altschuld and Brierley (9A) and Borrebaek (95) used high external concentrations of ATP to start their reactions and had low rates of ATP synthesis. Under these conditions Erickson-Viitanen et al. (92) observed a very low specific activity of creatine phosphate indicating that preferential access of MiMi-CK to the ATP generated by oxidative phosphorylation cannot be observed using the system of Altschuld and Brierley (66) or Borrebaek (95). The apparent increased activity of MiMi-CK in the space between the inner and outer mitochondrial membranes, shown by lower Km values for ATP (89 - 91). and increased rates of creatine phosphate production (89. 90), also results in an increased ADP concentration in the inner membrane space. This is shown by Moreadith and Jacobus (96) who measured the effect of increasing atractyloside concentrations on the rate of respiration in the presence and absence of 20 mg creatine. Unfortunately their results are difficult to interpret because the concentration which inhibited respiration by 50% is the same when either 20 mM creatine or hexokinase (coupled to agarose beads) and glucose are added (based on the percentage of oxygen consumption in the absence of added atractyloside). 17 However, a direct measurement of the nucleotide translocase Km for ADP shows that the Km value for ADP decreases from 13.2 uM to 2.9 u! in the presence of 50 m! creatine (97). This result is confirmed by Barbour gt El' (98) who also demonstrated that the exchange of matrix ADP with external ATP is slower when creatine is present, suggesting that active MiMi-CK creates a localized pool of ADP which does not equilibrate with the bulk solution as quickly as MiMi-CK turns over. The question of whether the outer membrane (87, 92, 93) or direct binding of MiMi-CK to the nucleotide translocase (88 - 90, 96, 99, 100, 101) is responsible for the decrease in the Km value for MgATP”2 is still unresolved. Bessman's group (87, 92, 93) argue that the presence of the outer membrane is necessary to create an unstirred layer which is responsible for maintaining a locally high ATP concentration in the inter membrane space. The contribution of the outer membrane to the coupling between the translocase and MiMi-CK is demonstrated by a higher apparent Km value for MgATP-2 when the outer mitochondrial membrane is present (in the presence of oligomycin A) and a failure to observe preferential access of MiMi-CK for the ATP generated by oxidative phosphorylation in the absence of the outer mitochondrial membrane (93). These results suggested to Erickson-Viitanen 35 al. (93) that the outer membrane is responsible for the preferred access of MiMi-CK for ATP derived from oxidative phosphorylation. The preferential access of MiMi-CK for ATP generated by oxidative phosphorylation may be the result of a diffusion barrier for MgATP—z which prevents its mixing with the solution MgATP‘Z (102 _ 10”). This barrier may result from either the creation of an unstirred layer around the IMM as is seen for whole cells (105), or the unequal ion distribution 18 which surrounds poly ionic surfaces (106). The outer mitochondrial membrane may act to stabilize these layers or may be a diffusional barrier to the movement of certain ions (107 - 110). Contrary to Erickson-Viitanen gt El. (92, 93), Saks and Jacobus and cowerkers (A0, 87 - 91, 96, 99, 100) maintain that preferential access results from the direct binding of MiMi-CK to the nucleotide translocase. Although the data do not support the direct transfer of ATP from the nucleotide translocase to MiMi-CK (111), they believe that the active sites of the two enzyme are close enough such that MiMi-CK can react with the ATP released by the nucleotide translocase before it diffuses into the surrounding solution. In this way, MiMi-CK will have an apparent lower Km value for MgATP-2 when preferentially coupled to oxidative phosphorylation. This substrate channelling has apparently been observed in large enzyme complexes which perform more than one reaction (A8). In an experiment designed to measure if the outer mitochondrial membrane could act as a diffusion barrier to ATP, Moreadith and Jacobus (96) measured the concentration of atractyloside required to inhibit liver mitochondrial respiration by 50% when respiration is initiated by UDP plus ATP or ADP. Since nucleoside diphosphokinase is present in the inner membrane space of liver mitochondria, if the outer mitochondrial membrane acts as a diffusion barrier then it should require more atractyloside to inhibit respiration when nucleoside diphosphokinase substrates (ATP and UDP) are used to initiate respiration as opposed to the situation when ADP alone is used. Their results apparently show that the amount of atractyloside required to inhibit 50% of respiration is identical for both cases, the respiration rates in the absence of 19 atractyloside are much lower when UDP plus ATP are used (96). This result, they conclude, shows that the direct binding of MiMi-CK to translocase, and not the outer mitochondrial membrane, is responsible for the observed lowering of the translocase Km value for ADP. However, Bessman agrues that the result suggests that the outer membrane is responsible for coupling because the concentration of atractyloside which inhibits respiration by 50% is clearly different when the curves are plotted as a percentage of respiration in the absence of atractyloside (87). The results of Hall and DeLuca (101) and Bennett giggl. (6), who examined creatine phosphate production as a function of increasing phosphate concentrations using mitochondria, show that the production of creatine phosphate decreases with increasing phosphate, suggesting that free enzyme in the inner membrane space is not as effective as bound enzyme in accepting ATP from the translocase. A more direct measure of the contribution of the outer mitochondrial membrane to preferential coupling is provided by experiments using mitoplasts. The Km values for MgATP-2 (at infinite creatine concentrations), measured in the presence and absence of oxidative phosphorylation, still show an apparent decrease when MiMi-CK is coupled to oxidative phosphorylation (100). However, Wenger gg_gl. (61) point out that neither Saks £2.2l' (100) nor Erickson-Viitanen gg gl. (93) indicate the percent of MiMi-CK bound to the inner mitochondrial membrane under the conditions of the experiment. Based on the concentration of inorganic phosphate required to release 50% of the MiMi- CK from mitoplasts (6) and the ionic strength of the assay media for both experiments, almost all of the enzyme should be free (58, 59, 61). The 20 significance of the coupling observed by Saks §£,E£- (100) is thus in question (61). MM-CK and Myosin ATPase Measuring tension development in cardiac fiber bundles (112) and skeletal muscle fiber bundles (113) which have been treated to increase the membrane permeability to small molecules such as creatine phosphate shows that tension develops at a faster rate and at 100 fold lower ATP concentrations in the presence of creatine phosphate (11A). This result suggested to Vesker and Kaplel'ko (112) that a locally high ATP concentration was generated around the myosin ATPase as a result of MM-CK action. In agreement with this hypothesis, Savabi gg gl. (113) report that the measured Km value of MM-CK for ATP is lower when contraction is initiated by creatine phosphate versus ADP (113) showing that ATP concentrations are higher near the myosin ATPase than in bulk solution. The actin activated magnesium ATPase activity is also significantly higher when creatine phosphate is added to the medium (5A). 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Saks, V.A., Ventura-Clapier, R., Huchua, Z.A., Preobrazhensky, A.N., and Emelin, I.V. (198A) Biochim. Biophys. Acta 803, 25A- 26A. Krause, S.M., and Jacobus, W.E. (1986) Biophys. 3. A9, Abstract # TPos-1. Chapter II Theory and Practical Application of Coupled Enzyme Reactions: One and Two Auxiliary Enzymes Published in Can. g. Biochem. Cell Biol. 62, 9A5-955 with permission of the copyright holder. Abstract An extended and practical set of equations which describe coupled enzyme reactions is presented. The mathematical treatment relies on two assumptions: (a) the rate of the primary enzyme reaction is constant and (b) the reverse reactions are negligible. The treatment leads to the development of new equations which relate the time required for the concentration of a reaction intermediate to reach a defined fraction of its steady-state concentration to the kinetic parameters of the enzymes when mutarotation of one of the intermediates does not occur. The new equations reduce to those previously derived when the steady state concentration of the intermediate is small compared to its Km value. A method for minimizing the cost of the two auxiliary enzyme system is also provided. Introduction Enzymologists commonly use two types of assay systems to monitor the course of an enzyme reaction: the end point assay and the continuous assay. For these assays to be correct, it is of paramount importance 29 30 that the steady state is achieved before the initial substrate concentration changes significantly. This is a trivial consideration when the product or substrate of the reaction can be monitored directly, but it becomes a major concern when auxiliary enzymes are used to detect the appearance of product. The equations relating the velocity of an enzyme to the rate of appearance of a detectable product resulting from subsequent (auxiliary) enzyme reactions were first developed by McClure (1) for systems involving two auxiliary enzymes. These were later extended by others (2- 7) to include multienzyme systems, and methods to minimize the cost of a two auxiliary enzyme system (3,A). The expressions, derived by several investigators (1-A), relate the velocity of the initial enzyme and the kinetic parameters of the auxiliary enzymes to the time required (tF, lag time) for the observed velocity to reach a defined fraction of the initial enzyme velocity. The derivations rely on the fulfillment of three assumptions: (a) the rate of product formation of the first enzyme is constant with respect to time (dv1/dt - 0), (b) the steady state concentration(s) of the intermediate(s) is (are) much less that the apparent Michaelis constant for the subsequent auxiliary enzyme(s), and (c) only initial velocity conditions are considered (the reverse reactions are negligible). Whereas none of the above conditions are necessary for the development of a generalized theory for coupled enzyme reactions, the first and third conditions are easily achieved by selecting appropriate substrate concentrations. These expressions allow calculation of the amount of coupling enzyme(s) needed to shorten the lag time so that the steady- state condition may be achieved while initial velocity conditions still 31 apply (d[ES]/dt - 0). Other investigators (6,7) recently developed expressions to describe the time course of coupled enzyme reactions when the intermediate concentration is greater than 0.01 Km of the coupling enzyme. Easterby's (6) elegant analysis in terms of the transition time, T, (see Figure II-1), although theoretically sound, is not practical as it does not define the time taken for the rate of the reaction to achieve a defined fraction of the steady state rate. Furthermore, the practical value, tF (the lag time), is not related to T by a simple mathematical expression: in order to obtain t it is necessary to solve a system of F’ differential equations. For the case involving two coupling enzymes, the differential equations can only be solved by numerical methods. The purpose of this paper is two fold: (a) to provide a simple method for calculating the amount of coupling enzymes required for a valid assay when the concentration of the intermediate(s) is greater than 0.01 Km and (b) to provide a method for minimizing the amount and cost of the coupling enzymes required for the two coupling enzyme system. Theory One Auxiliary Eggyme The general form of a coupled enzyme reaction is as follows: Scheme 1 3 V1 ) B V2 > P primary fluxiliary\ enzyme Q enzyme R (I) (II) where v is the rate of the enzyme under study and v 1 is the rate of the 2 auxiliary enzyme. Since the auxiliary enzyme usually catalyses a two Figure II-1: Figure II-2: 32 Relationship between product concentration and time for a theoretical enzyme assay. The solid line designated with P represents the time course for the appearance of product: the dashed line on the P curve is an extrapolation from the steady state rate for the appearance of product to the time axis. The intersection of the extrapolated dashed line with the time axis defines Easterby's I value (6). The solid line designated B represents the time course for the appearance of the intermediate B: the dashed line from the B curve represents the concentration of B in the steady state. The time at which the concentration of the intermediate is 99% of its steady state value is t99. Theoretical prediction from Equation II-7. Note v = v s for all values of v < V . The figure presupposes measurements at infinite time. 33 [P] CODC. [B] vobs 3A substrate reaction, it is represented above as B + Q ---> P + R where R and Q are generally the oxidized or reduced forms of the pyridine nucleotides, NADP(H) or NAD(H). We will begin by assuming that the first reaction obeys the initial velocity condition in the derivation of the Briggs-Haldane steady-state equation: [II-1] d[E1S]/dt = O and v1 = V1[S]/(KS + [3]) where [E18] is the enzyme-substrate complex of enzyme 1, v1 is the velocity of enzyme I, V is the Km for is the V for enzyme I and K max 1 S substrate S. The rate of disappearance of B is given by: [II-2] -d[B]/dt - V2[B]/(KB + [8]) where V2 is the apparent maximal velocity for enzyme II at finite concentrations of the second substrate (Q) and KB is the Michaelis constant for B. The relationship between V and the velocity of enzyme 2 II at infinite concentrations of B and Q (V2 max) is given by Equation II-3: [II-3] v2 . 1:2,"!ax [QJ/(KQ + [01) where K - - Q is the apparent Michael1s constant for Q. V2 is then expressed in concentration terms, units per assay or units per milliliter; its value may be changed either by adding more auxiliary enzyme to the assay 35 or by increasing the concentration of the substrate Q as shown by Equation II-3. The observed steady state velocity of a coupled enzyme reaction is simply the velocity of enzyme II: [II-A] vobs = v2 [BJSS/(KB + [8133) When the steady state concentration of B is less than 0.01 K Equation B! II-A simplifies to the relationship derived by others (1-A). [II-5] vobs g v2 [Bjss/KB In order to calculate the elapsed time before the onset of steady state, we must first obtain an expression for formation of B in terms of time (t). The expression shown in Equation II-6 is the difference between the rate of formation of B (v1) and the rate of its destruction (VZEBJ/(KB + [B])) [II-6] d[B]/dt = v1 - v2 [BJ/(KB + [8]) At steady state, when d[B]/dt = 0, Equations II-6 and II-A predict: [II-7] v1 = vobs = d[P]/dt - d[R]/dt Equation II-7 implies that when the reaction enters steady state, the observed rate will be that of enzyme I. This is depicted graphically in Figure II-2. A plot of yobs versus v1, gives a straight line with a 36 slope of 1 until the rate of enzyme I is equal to the rate of enzyme II; at this point and at all values of v1 in excess of V2, the observed rate equals V2. By holding v and V constant with respect to time, Equation 1 2 II-6 gives Equation II-8. (KB + [8]) dEB] [II-8] = fdt VIKB + [B](v1 - V2) Integration of Equation II-8 assuming [B] = 0 when t = 0 gives the complete equation for [B] at any time t [II-9] (KBv 1 - ¢[B])exp(¢[B]/KBV2) = KBviexp(-t¢2/KBV2) Where ¢ - V2 - v1. When v1 << V2 and [B] << KB, Equation II-9 gives Equation II-10, the equation obtained by others (1-A). K13V1 [II-10] [B] - . [1 - exp(-tV2/KB)] V2 Following McClure (1) we define: [II-11] FB - [B]/[B]SS where [B] is the concentration of the intermediate, B, at any time t and [B]SS is the concentration of B in the steady state given by: [II-12] [B]SS - V1KB/(V2 - v1) = v1KB/¢ Combining Equations II-9. II-11 and II-12 we obtain 37 -K B [II-13] t = 7EFBV FB ¢ + V21n(1-FB)] 1 Where tFB is the lag time before B reaches a defined fraction of its steady state concentration as given by Equation II-11. If v1 << V2, we obtain the previously derived equation (see reference 1) [II-1A] tFB = -;- ln(1-FB) 2 Figure II-3 shows the relationship between the lag time (tFB, with F - 0.99) calculated from Equation II-13 and the ratio v B /V2. Note 1 that as the ratio v1/V2 increases above 0.1, the time required for B to achieve 99% of its steady state concentration increases dramatically. If FB is selected as 0.95. the actual lag times are shorter but follow the same trends as seen forFB . 0.99. Thus v1/V2 should be less than 0.1 if the steady state rate is to be reached under conditions when the steady state assumptions are still valid. This is particularly important when kinetic data for enzyme I are examined at concentrations of substrate below the Km value. The tFB value in Equation II-13 reflects the time required for the intermediate concentration to achieve a defined fraction (FB) of its steady state concentration ([8188). This is in contrast to the earlier equations developed by Storer and Cornish-Bowden (5) and Easterby (6) where the lag time (tFB) reflects the time taken for the observed velocity to achieve a defined fraction of the steady state velocity. A comparison of Equation II-13 and the relationship given by Storer and Cornish-Bowden (5) shows that the calculated lag times differ by 3% at 38 most. However, in contrast to the expression developed by Storer and Cornish-Bowden, Equation II-13 allows the direct calculation of V2 or v when either v1 or V2, tFB and KB are known. Rearranging Equation II-13 1 in terms of V2 we obtain: 2 1 + FBKBv1/tFB - O V 2 2 + V2[ln(1-FB)KB/tFB - 2V1] + V Thus the amount of enzyme II (V2 in units) required for a defined lag time (tFB) is easily calculated. Two Auxiliary Enzymes The general form of a two auxiliary enzyme reaction can be written as follows: Scheme 2 v v v A 1 > B 2 > C 3 > P primary auxiliary /auxiliary enzyme enzyme 1 Q enzyme 2 R (I) (II) (III) Following the treatment developed for the one auxiliary enzyme system, we obtain [II-15] d[C]/dt = V2[B]/(KB + [3]) = V3[C]/(KC + [C]) where KC is the Km for C and V3 is the apparent maximal velocity of auxiliary enzyme III at finite substrate concentrations, or [II-16] v3 = v3,max [Ql/(KQ + [0]) Figure II-3: Figure II-A: 39 Theoretical relationship between log(v /V ) and the time required for B to reach 99% of its steady state value. The t9 values were calculated from Equation II-13. Each curve Represents a different value of V2 as shown in the figure. Relationship between t and log(v1/V3). 99 The values of t were obtained with KINFIT A (8) using Equation II-6 egg II-15. KB = 10 pg, V - 100 units, K - 10 pg. 0, v = so units; o,V ~= 66.6667un1ts: 1,v =- 69 units. A re ction volume of mL is assumed. The 3olid lineS'are calculated using Equation II-22. The dashed lines are from Equation II-21a. ' 40 199 (min) 0.5~ rvzesoo o ‘ i l ‘ 1 -6 -5 -4 '3 ’2 4 o Log(v,/V2) 1.5 1 ' ' i I I / / 1.o - ’ _ / 199(m1n) 1 \ \ \ 0.5 - A1 where V3 max is the velocity at infinite substrate concentration and KQ 9 is the Michaelis constant for Q. When d[CJ/dt - 0 (i.e. [B] = [B]SS and [C] = [CJSS) Equations II-6 and II-15 predict that [II-17] vobs = V3[C]SS/(KC + [C]88) - V1 and thus, in agreement with Easterby (8), we obtain the general relationship II-18 I . = v K./ V. - v ['1 [JJSS 1‘3 ) where [I]js is the steady state concentration of the jth intermediate, 3 and Kj and Vj are the Km and Vmax for the enzyme following the appearance of Ij. Because Equation II-15 cannot be integrated by conventional methods, an exact solution for the intermediate [C], and consequently, an exact value of tFC cannot be obtained as was done so conveniently for the case of one auxiliary enzyme. We, therefore, developed equations to approximate the concentration of C at any time t. To do this, equations to approximate the concentration of the intermediate for the one auxiliary enzyme case were first developed. These equations were obtained by rewriting Equation II-6 as two separate equations, each defining an upper and lower limit for the concentration of B over the time course of the experiment. Using the solutions to these equations (see Appendix), it was possible to obtain an approximate solution for the two auxiliary enzyme system described by Equation II-15. As with the one auxiliary enzyme system, we obtain two equations which define an upper A2 and lower limit for the concentration of the intermediate ([CJ). The lower limit is defined by Equation II-19a and the upper limit by Equation II-19b: [II-19a] V VZKBV K o [C_] = [C]33[-— + exp(-V2t/KB) - ----—exp(-V3t/KC)] V3 ¢(KCV2 - KBV3) ¢KC - 1K8 [II-19b] V 6K [0+] = [CJSSE1 + ¢K - w exp(-¢t/KB) - ¢K _ wK exp (-Vt/KC)] ch. 0 B where V = V3 - v1. When V3 >> v1 and V2 >> v1, we can obtain, from Equations II-19a and II-19b, the equation for the pseudo-first order case originally derived by McClure (1): v v k 2 [II-20] [c] . -— — , [exp(-k t ) + -exp(-k t )1 33 k3 (k3 _ k2) 2 FC k3 3 FC where k2 - V2/KB and k3 tFC+ and th_ values obtained from Equations II-19a and II-19b when o/K - O.A8 Y/KC. The calculations reveal that the tFC- value is the better estimate and we therefore provide the following equations which apply - V3/KC. Table II-1 shows a comparison of the B when the indicated conditions are valid. When o/KB Z_27/KC: [II-21a] t = -KCln[(1-FC)(1 — vKB/6KC)]/v FC- When 2¢/KB S V/KC: A3 [II-21b] t _ - -KBln[(1-FC)(1 - 6K PC /vKB)]/¢ C A more precise estimate of t can be obtained by a numerical FC solution of Equation II-22, an empirical equation developed from McClure's equation 10 (1) (our Equation II-20). We reasoned that equations for the two enzyme case could be developed which have the same relationship to each other as Equation II-13 and II-1A have for the one enzyme case. Starting with Equation II-20, which was developed for the two enzyme system when V and V >> v1, and substituting c/v for V and 2 3 2 2 WV3 for V3 and multiplying the term (1-FC) by exp(FCv1/V3), we obtain the following equation: [II-22] ¢ZKCV3 2 OZKCV3 2 (1- -§--- )(1-FC)exp(FCv1/V3) = exp(-tFC¢ /V2KB) - -§--- exp(-tFCV /V3KC) W KBVZ . V KBV2 Arguments supporting the usefulness of Equation II-22 are: (a) the variable (in this case tFC) shows a symmetrical dependence on V K 2’ 8’ V3 and KC (this is expected as lowering either V2 or V3 will affect the lag time) (b) the expression contains the term FC only because we are looking for the time taken for the final intermediate in the reaction sequence to reach a defined fraction of its steady state concentration. At this time, F8 is very close to 1.0. Table II-1 shows a comparison of the calculated concentrations of C using Equations II-21 and II-22. Equation II-22 gives a better estimate but a computer solution is necessary (see Appendix). An analysis of Equation II-22 over a wide range or V2. V3, KB and KC values shows that the error in calculating [C] is usually less than 5% for any value of V2, V3, KB and KC for which AA Table II-1 Comparison of t Values Calculated by Various Methods for the Two Auxiliary Enzyme System. FC tFC- %Error tFC+ %Error tFC %Error tFC (emp) (exact) (min) (min) (min) (min) 0.901172 1.6A235 -3.A 1.88333 10.8 1.75903 3.5 1.7 0.915877 1.73185 -3.8 2.0625 1A.6 1.86035 3.A 1.8 0.928A1 1.821A8 -A.1 2.29137 20g6 1.960A5 3.2 1.9 0;939087 1.9112 -A.A 2.63252 31.6 2.06005 3.0 2.0 0.9A8179 2.00101 -A.7 3.5278 68.6 2;1606A 2.9 2.1 0;955918 2.09087 -5;0 ‘ * -‘ 2.26000 2:8 2.2 0.962505 2.18077 -5.2 * - 2.36000 2.6 2.3 0.968109 2.27071 -5.A * - 2.A6100 2.5 2.A 0.972877 2.36068 -5.6 * - 2.56100 2.A 2.5 0.976933 2.A5066 -5.7 * - 2.66110 2.A 2.6 0.980382 2.5A066 -5.9 * - 2.75900 2.2 2.7 0.983317 2.63066 -6.0 * - 2.86130 2.2 2.8 0.985812 2.72068 -6.2 * - 2.99590 2.0 2.9 0:98793A 2;81069 -6.3 * - 3;0566O 1.9 3.0 0.989739 2.90071 -6;A * - 3.15A3O 1.8 3.1 0.99127A 2.99072 -6.5 * - 3.26270 1.9 3.2 Note: tF _, t + and tF (emp) were calculated using Equations II-21, II-19a ans II-EE, respecEively. t (exact) was obtained from a Runge- Kutta numerical integration of Scheme 2. The FC values were obtained from the t (exact) solution using F - [CJ/[C] and [C] - 0.263 uM. Other valugg are as follows: v1 = 1 uM/min, V is10 uM/miR§ V * ‘-' 20 uM/min, KB - 5 ufl, KC - 5 pg. The asterisR (*) denotes thé case when FC Z V/V3. ' N5 oZ/VéKB ¢ 92/V3KC. It should be noted, however, that the estimate of t PC is always greater than the true value of tFC’ and improves as FC increases. Figure II-fl shows the t values generated by Equations II-6 and 99 II-15 (symbols), Equation II-21a (dashed lines) and Equation II-22 (solid lines). The t values given by the symbols in Figure II-A are exact 99 values obtained by numerical integration of Equations II-6 and II-15. The estimate obtained from Equation II-22 is always better than the tFC- analysis. When Y/K - Q/KB, Equation II-19a or II-22 must be solved C numerically for t When Equation II-19a is used, the following value PC? of [C+] must be used: [II-23] [0+] = F + [c] C $3 The reader is cautioned against the use of an equation of the form [II-2“] tFC = KCln(1-FC)/W which can be derived from Equations II-19a and II-19b (see Appendix) or 22 when O/KB - V/KC: the results obtained from Equation II-ZU are then in error by at least 301 as compared to the true value obtained by numerical integration (8). Therefore, the curves corresponding to V2 - V3 - 100 and K - K =- 10 in Figure II-u are calculated by assuming V - 99 and C B solving Equation II-19a or II-22 for t 3 FC by a method similar to that shown in the Appendix. It should also be noted that interchanging V2 for V3 and KB for KC results in the same t99 value. This is reflected by Equations II-19a and II-19b and Equation II-22. Thus Figure II-u would A6 be identical, if we plotted t99 versus log(vI/V3) instead of versus log(v1/V2). Minimum Values of V9 and V9: (_ 3— For a reaction involving two auxiliary enzymes, the value of tFC can be obtained from the above analysis only when both V2 and V3 are known. Furthermore, Equations II-19a and II-19b and II-22 show that a wide range of V2 and V3 values can be obtained for a single tFC value. Therefore, in order to select a single value of V and V 2 3, it is useful to derive a function which will minimize the concentration of each coupling enzyme required to give a selected tFC value at the minimum cost. For a reaction involving two auxiliary enzymes, the total cost per assay is simply the sum of the concentration of the enzyme in units (V2 or V3) multiplied by the price per unit (P2 or P3). [II-25] Cost = P2V2 + P3V3 Because the functions which describe the lag time are not explicit for V2 or V , we use the transition time (1) function (see Figure lI-1). From 3 Easterby (6) we know that for a system of coupling enzymes. [II-26] T = {IK‘j/(V‘j - V1) Thus for two enzymes: [II—27] : = KB/¢ + KC/v Figure II-5: Figure II-6: M7 Graphical representation of the cost minimization technique. The three curves, generated by Equation II-28, represent the relationship between V and V for various values of I. The slope of the straight solid line is defined by 6 from Equation II-29 with P /P = 3 and 1 - 0.8 min. The dotted lines are extrapolations3of the point of intersection of the functions from Equation II-28b and Equation II-29 to the axis. A reaction volume of 1 mL is assumed. Relationship between tFC and T. The value of t was calculated using Equation II-22 and T (min) was obtained from Equation II-27. The solid lines represent the line of best fit through the points. 0, relationship between t99 and T; 0, relationship between t and I. 90 I50 V5 (00".) 'F (min) 48 50-} It ' 2.0 ’6 -O.8 L. -... 50 100 V2 (00"., I50 1.5 H9 which gives after rearrangement [II-28a] V v1 + KEV/(IV - KC) [II-28a] V v + KC0/(10 - KB) 1 Substituting Equations II-28a and II-28b into Equation II-25, taking the derivative of cost with respect to V and V (holding 1 constant), and 2 3 setting dCost/dV2 = 0 and dCost/dV3 = 0, we obtain the following relationship: v PP(1V+)+P(PPKK)1/2 2 3 2 1 KB 3 2 3 C B [II-29] 6 = - = - 1/2 v3 P2P3(rv1 + KC) + P2(P2P3KCKB) where 6 is the slope of the line of minimum cost as depicted in Figure II-5. The minimum values of V2 and V3 required to give a Specified 1 value occur at the point of intersection between Equation II-28a or II- 28b and II-29. This point is given by Equation II-30: 2 1/2 R1 + (32 + GKBKC) [II-30] V = 2 2T where R1 = Iv1 + KB + 6(KC + 1V1) R2 = 1V1 + KB - 6(KC + TV1) and V3 = V2/6. Figure II-5 shows a plot of V2 versus V3 derived from Equation II-28b. Note that the value of 6 differs for each I value as shown in Equation II-29. Although the functions described above minimize the I value (and consequently are useful for minimizing V2 and V3), the I value has no 50 99 and t95 values (obtained with Equation II-22) versus 1 (calculated from Equation practical use. However, a plot given in Figure II-6 of t II-27) demonstrates that an approximate linear relationship between the two parameters can be obtained. Thus 1 can be calculated from the tFC values by t a ".281 and t . 2.791. 99 90 The relationships between t99, t90 and 1 are obtained from a least square analysis of the points shown in Figure II-6. It must be stated that the actual values of tF/T can vary considerably from the above relationships and that these numbers serve only as an approximate guide. Once apprOpriate values of V2 and V3 are obtained, one must return to Equations II-19a and II-19b or II-22 to obtain an exact lag time. DISCUSSION To set up a successful coupled enzyme reaction, one needs to know the amount of coupling enzyme to add to give as short a lag time as possible consistent with a reasonable cost. Most investigators simply add excess coupling enzyme to an assay and confirm that the observed rate is directly proportional to the amount of primary enzyme added. While this is satisfactory for a limited number of assays, it is not cost effective. A short lag time is desirable for at least two reasons: (a) the shorter the lag time, the lower the steady state concentration of product intermediates and the smaller the affect, if product inhibits, and (b) the shorter the lag time, the less substrate is consumed before the steady state rate is observed. However, the shorter the lag time, the more coupling enzyme required. Therefore, it is important to be able to 51 calculate the amount of coupling enzyme to add for an assumed but defined reasonable lag time. Previous theoretical treatments of coupled enzyme reactions relied on three assunptions: (a) the rate of the primary enzyme is constant, (b) the reverse reactions are negligible and (0) sufficient coupling enzyme is added so that the steady state concentration(s) of intermediate(s) is (are) small or (KB + [8]) = K The first assumption 8'. forms the basis of all initial velocity measurements; adding a coupling enzyme would not be expected to invalidate it. The second assumption is considered valid because coupling enzymes usually displace the equilibrium of the primary enzyme by removing substrate. Because the third assumption restricts the usefulness of available equations, the equations in this paper were developed with the assumption that (KB + [B]) s KB. Previous investigators also assumed that the coupling enzymes obey Michaelis-Menten kinetics even though the concentration of intermediates in the coupling system increases from zero to some steady state value with time: Michaelis-Menten kinetics assumes d[SJ/dt - 0. To test this assumption, Equation II—6 and the differential equations that describe the reactions in Scheme 3 were integrated numerically and found to give identical values for vobs at FB > 0.9 (unpublished observations). Thus Michaelis-Menten kinetics adequately describe coupled enzyme systems. Scheme 3 52 The equations for calculating the units of coupling enzyme(s) necessary to produce a defined lag time, tF’ when v is known and when 1 (KB + [B]ss) a KB, are presented for both the one coupling enzyme system (Equation 11*13) and for the two coupling enzyme system (Equation II~19a and II-19b and II-22). The difficulties encountered when incorrect lag times are calculated, when it is assumed that (KB + [B]SS) = KB, are discussed in more detail in the accompanying paper (9). A method for minimizing the total concentration of V2 and V3, and thus the cost of the assay in the two coupling enzyme system is also provided. To proceed, the investigator must first select a desired lag time, t C' and them F estimate a transition time, 1, from the relationship shown in Figure II- 6. The minimum concentrations of V2 and V3 are calculated using Equations II-29 and II-30 and the selected I value. The correct lag time must then be recalculated using Equation II-19a or II-19b or equation II- 22 and the new values of V2 and V3. Acknowledgment S.B. wishes to express his thanks to Dr. P. Nicholls for helpful discussions during the initial stages of this paper. References 1. McClure, W.E. (1969), Biochem.,8, 2782-2786. 2. Easterby, T.S. (1973), Biochim. Biophys. Acta, 223, 552-558. 3. Cleland, W.W. (1979), Anal. Biochem., 22, 1u2—1us. A. Garcia-Carmona, F., Carcia-Canovas, F. and Lozano, J.A. (1981), Anal. Biochem., 113, 286-291. 5. Storer, A.C. and Cornish-Bowden, A. (197”), Biochem. J., 1N1, 205- 53 209. 6. Easterby, T.S. (1981), Biochem. J., 122, 155-161. 7. Takagahara, l., Yamauti, J., Fujii, K., Yamashita, J. and Horio, T. (1983), J. Biochem., 23, 11A5-1157. 8. Dye, J.L. and Nicely, V.A. (1971), J. Chem. Educ., £8, NA3-NA8. 9. Brooks, S.P.J., Espinola, T. and Suelter. C.H. (198“), Can. J. Biochem. Cell Biol., 62, 956-963. Appendi 1: One Auxiliary Enzyme The equations which approximate the concentration of B over the time course of the experiment were obtained by rewriting Equation II-6 as two separate equations [II-Ala] d[B+]/dt v - k+[B+] 1 v. - k_[B_] [II-A1b] d[B_]/dt 1 where [8+] and [B_] are the upper and lower estimates of the true concentration of [B]. [8+] and [B_] are defined by the appropriate choice of the constants k+ and k_. Because the constant k+ must satisfy the condition [8+] 3 [B], Equation II-A1a dictates that k+ must be equal to the smallest value for the expression V2/(KB + [8]) (see Equation II- 6) for any time t or 5N + [II-A23] k = V2/(KB + [8138) = d>/KB Conversly k_ must be equal to the largest value of VZ/(KB + [8]) or: [II-A2b] k_ = V2/KB (McClure's (1) assumption) It is readily seen that with these assignments, the condition k+ _<_ V2/(KB + [8]) g k_ is satisfied and consequently [II-A3] d[B_J/dt i d[BJ/dt 5 d[B+]/dt is always true. Integration of Equations II-A1a and II-A1b gives two equations of the general type: [II-AA] [B] = v1/k + Z exp(-kt) 1 where the value of Z1 is obtained by selecting the initial (t=0) condition. This initial condition must satisfy the imposed criterion that tFB+/- is the best estimate of the true tFB value. Selecting [B+Jt=0 = O we obtain 21+ = v1KB/0, and Equation II-A1a becomes: V1KB [IL-Asa] [8+] = (1 “ €Xp(-¢t/KB)) 0 If we choose [B__]t=0 = 0, Equation II-A1b becomes McClures's 55 equation 2 (our Equation II-10). However a better estimate of tFB can be obtained if we let 2 - Z1+ (i.e. [B_] 1_ < O). This gives Equation II- t=0 A5. v K v K [II-A5b] [B_] = exp(-V2t/KB) V2 ¢ and it follows from Equation II-A5a and b that for any time t v K v K v K 1 B(1-exp(-> V], are also shown for comparison. The best estimates of [B] and t are derived from the B+ analysis. The values of t FB FB- Figure II-A1: 56 a.Graph of [B]/KB versus reaction time calculated by three methods. 0, [B_]lK (see Equation II-A5b); o, [8+J/KB II-A5a) and A,[B]/KB from Equation II-11. v V2 - 1O uM/min. (see Equation 1 - 1 uM/min, b.Plot of tFB versus FB calculated from various equations. Solid line (tFB exact) from Equation II-13. The dashed line (tF _) is from Equation II-A6. The dotted-dashed line is from Equation lI-1M (McClure's estimation (1)). v1 = 1 uM/min, V2 = 10 uM/min. ' 57 A "‘ [essJ/Ka t--_._/_-__ 010 - ' [Bl/K3 *8 (min) (105 o -oxn; 1(min) 58 (dashed line, Figure II-A1b) are always slightly low due to the fact that the value of [B] calculated by the B+ analysis is always greater that the true value of [B] and consequently the calculated value of [B] achieves the true steady state concentration faster. The values of tFB+ are always infinite since the concentration of B defined by Equation II-A5, and the parameters indicated in the figure legend, never achieve [B]SS; (<1>/V2 - F8) is always less than or equal to zero for F 3 0.9. B Two Auxiliary Enzymes Having defined the values k+ and k_ for the one enzyme case we can proceed to estimate the value of t for the two enzyme case by F8 rewriting Equation II-15 as: [II-A7a] d[C+]/dt = k+[B+] - r+[C+] [II-A7b] d[C_]/dt = k_[B_] — r_[C_] with k_ and k+ defined as above. Following the above rationale, we choose r+ = WKC and r_ - V3/KC where W - V3 - v1. Integration of Equation II-A7 gives an equation of the general form: v1 kZ1 [II-A8] [C] - - + r (k-r) exp(-tk) + Z exp(-tr) 2 By selecting an appropriate initial or final condition such that the tFC estimate is close to the true tFC value, Equations II-19a and II-b are obtained (refer to the text). 59 Program minimum the mini the 100 200 300 A00 500 600 700 800 900 1000 1100 1200 1300 1A00 1500 1600 1700 1800 1900 2000 2100 2200 2300 ZAOO 2500 2600 2700 2800 2900 3000 3100 3200 3300 3A00 3500 3600 3700 3800 A program written in the BASIC language useful for calculating units of enzyme II and III required to give a valid assay at a mum cost for a two enzyme coupled assay system given by scheme 2, and lag time before the steady state is achieved (see Equation II-22). PRINT "THIS IS PROGRAM MINIMUM. IT WILL CALCULATE THE MINIMUM AMOUNT" PRINT "OF ENZYMES II AND III TO ADD IN ORDER TO MINIMIZE THE TOTAL COST" PRINT "OF A COUPLED ENZYME ASSAY. IT CAN ALSO CALCULATE THE LAG TIME FOR" ' PRINT "GIVEN VALUES OF V2 AND V3." PRINT ' REM**************************************************************** REM REM THE PROGRAM FIRST ASKS YOU TO SELECT OPTION A OR B REM OPTION A CALCULATES THE MINIMUM AMOUNT OF ENZYMES II AND III TO REM ADD WHEN GIVEN AN INITIAL ESTIMATE OF THE LAG TIME. IT THEN REM RECALCUCLATES AN APPROXIMATE LAG TIME BASED ON EQUATION II-22 REM REM*************************************************************** PRINT "WHAT DO YOU wANT TO DO ?" PRINT " ——" PRINT "(A) CALCULATE THE MINIMUM COST AND LAC TIME." PRINT - PRINT "(3) CALCULATE THE LAG TIME ONLY. "; INPUT A$ ' IF A$ <> "A" AND A$ <> "B" THEN 1A00 REM***I******§§*************************************************** REM REM THE PROGRAM NEXT ASKS you To ENTER THE FOLLOWING VARIABLES: REM KB AND KC (AS K2 AND K3). FC (AS F), v1 (AS A) REM - REM************************************************************** PRINT PRINT "ENTER KB, KC (MICROMOLAR) "; INPUT K2,K3 PRINT PRINT "ENTER FC "; INPUT F IF F <- 0 0R F >= 1 THEN 3100 PRINT PRINT "ENTER THE VELOCITY OF ENZYME I (MICROMoLAR/MIN) "; INPUT A REM********l****§************************************************ REM 3900 A000 A100 A200 A300 AAOO A500 A600 A700 A800 A900 5000 5100 5200 5300 5A00 5500 5600 5700 5800 5900 6000 6100 6200 6300 6A00 6500 6600 6700 6800 6900 7000 7100 7200 7300 7A00 7500 7600 7700 7800 7900 8000 8100 8200 8300 8A00 8500 8600 8700 8800 8900 REM REM 60 IF OPTION A IS SELECTED , THE PROGRAM JUMPS TO SUBROUTINE MINIMUM TO CALCULATE THE VALUES OF V2 AND V3. REM IF OPTION B IS SELECTED, THE PROGRAM JUMPS TO SUBROUTINE REM REM VELOCITY AND ASKS FOR V2 AND V3 REM****************************************************§********* IF A$-"A" THEN GOSUB 8800 IF A$-"B" THEN GOSUB 13000 REM*********§******&***************Niifiiiiiiiiiiiiiiiiili§******* REM REM NEXT THE PROGRAM CALCULATES THE VALUE OF TIME (T) BY GUESSING REM ITS VALUE AND CHECKING IF THE VALUE T1 IS ZEROED. REM IF IT IS NOT, THE PROGRAM RETURNS TO LINE 6300. REM THE VALUE T1 IS CALCULATED BY SUBTRACTING THE LEFT AND RIGHT REM SIDES OF THE EQUATION IN THE SUBROUTINE. WHEN LS-RS THE VALUE REM OF T HAS BEEN FOUND AND T1=O. REM Q IS A POINTER SET ONLY IF THE INEQUALITY IN LINE 7600 NEEDS REM TO BE REVERSED. THIS MAY OCCUR IF THE EQUATION IS SOLVED IN A REM DIFFERENT MANNER DEPENDING ON THE PARAMETERS. REM ' REM*********************************************§****§*******H}! Q-O T55 E-LOG(T/2)/LOG(2)+1 GOSUB 11500 T1-INT(T1*10000) IF T<9o 99 AND T>O. .01 THEN 6800 Q-Q+1. 1 GOTO 6100 IF Q<2 THEN 7200 PRINT PRINT "THE PROBLEM IS UNSOLVABLE 1!" GOTO 13900 IF T1<>O THEN 7600 PRINT PRINT "THE TIME REQUIRED TO REACH ";F;" STEADY STATE IS: ":T;" MINUTES." GOTO 13900 IF Q<1 AND T1<0 THEN 7900 IF Q>1 AND T1>O THEN 7900 T-O *T E-E-1 T - ABS(2“E+T) GOTO 6300 REM*****§***********************************************§****** REM REM THIS IS SUBROUTINE MINIMUM. THIS SUBROUTINE USES EQUATION REM REM II- 29 T0 CALCULATE V2 AND V3 MINIMUM. REM*********§*****************************§******************** PRINT PRINT "ENTER THE APPROX. VALUE OF T";F*100;" DESIRED (MIN.) "; 9000 9100 9200 9300 9A00 9500 9600 9700 9800 9900 10000 10100 10200 10300 10A00 10500 10600 10700 10800 10900 11000 11100 11200 11300 11A00 11500 11600 11700 11800 11900 12000 12100 12200 12300 12AOO 12500 12600 12700 12800 12900 13000 13100 13200 13300 13A00 13500 13600 13700 13800 13900 61 INPUT B J-16.5556*F-12.11 B1hB/J - PRINT PRINT "ENTER THE COST OF ENZYME II, III (COST/UNIT) "; INPUT P2,P3 A6-P3*(P2*(B1*A+K2)+SQR(P2*P3*K2*K3)) A7-P2*(P3*(B1*A+K3)+SQR(P2*P3*K2*K3)) A8=A6/A7 * R1-A*B1+K2+A8*(K3+B1*A) R2-V1*B1+K2-A8*(K3+B1*V1) V2=(R1+SQR(R2“2+K2*A8*K3))/(2*B1) V3=V2/A8 PRINT PRINT "THE MINIMUM VALUES OF V2 AND V3 ARE: PRINT PRINT "V2 8 ";V2, "V3 - ";V3 RETURN REM********************************************************** REM REM THIS IS SUBROUTINE EQUATION. THE EQUATION YOU WISH TO REM SOLVE GOES HERE. THE VALUE T1 MUST BE RETURNED REM SO THAT THE PROGRAM WILL FIND THE CORRECT LAG TIME (T). REM REM§§¥§§§§§§§¥§§¥¥§§§***********ii§§¥fli¥§i§i§ii¥§************ CI-(V2-A)‘2 C2-(V3-A)“2 IF (C1/V2*K2) <> (C2/V3*K3) THEN 12000 V3-V3-V3/1OO GOTO 11500 L=EXP(—T*C1/(V2*K2))-C1*K3*V3/(C2*K2*V2)*EXP(-T*C2/(V3*K3)) R-(1-C1*K3*V3/(C2*K2*V2))*(1-F)*EXP(F*A/V3) T1-L-R‘ RETURN REM**§***********************************!********!*******iii REM REM THIS IS SUBROUTINE VELOCITY. REM THIS IS THE INPUT FOR V2 AND V3 IF OPTION B WAS CHOSEN REM REM*********************************************************** PRINT PRINT "ENTER V2, V3 (MICROMOLAR/MIN) "; INPUT V2,V3 RETURN REM********************************************************** REM REM THIS IS THE END OF THE PROGRAM REM REM**1-*************§***************************************** END Chapter III Theory and Practical Application of Coupled Enzyme Systems: One and Two Coupling Enzymes with Mutarotation of an Intermediate Published in Can. 3. Biochem. Cell Biol. 62, 96A-971 with permission of the copyright holder. Abstract This paper provides equations to calculate the elasped time before the concentration of the final intermediate, in a sequence of coupled enzymatic reactions, achieves a defined fraction of its steady state concentration when one of the intermediates undergoes mutarotation. The equations can be used to predict lag times for systems involving one coupling enzyme, as is the case when hexokinase or phosphoglucomutase activity is monitored using glucose 6-phosphate dehydrogenase as the auxiliary enzyme, or for systems of two coupling enzymes, as is the case when the activity of enzymes producting ATP (such as creatine kinase) are monitored by coupling the production of ATP to hexokinase and glucose 6- phosphate dehydrogenase. The theoretical aspects of the assay have been verified using hexokinase (as the primary enzyme) and glucose 6-phosphate dehydrogenase (as the coupling enzyme). A method of cost minimization, based on the above relationships, is also provided. 62 63 Introduction It is common practice for enzymologists to use auxiliary enzymes as a means to assay the activity of an enzyme when its product is not detectable by conventional techniques. When coupling enzymes are employed, the observed velocity is not constant over the time course of an experiment: it increases until it approximates the rate of the primary enzyme. The final steady state rate is achieved after a defined lag time (see Figure III-1). It is important that the experimenter be capable of calculating this period, because the observed velocity does not accurately reflect the rate of the primary enzyme before a lag period has elapsed. One of the most common coupling enzyme systems is the hexokinase- glucose 6-phosphate dehydrogenase system which is often used to monitor ATP production. The product of the hexokinase reaction, glucose 6- phosphate, undergoes mutarotation at the carbon one position giving a mixture of a and B enantiomers (1). Because glucose 6-phosphate dehydrogenase reacts only with the B-enantiomer, the observed lag time (tgg, Figure III-1) will depend not only on the kinetic constants of the coupling enzymes, but on the rate of interconversion of the a and 8 enantiomers as well (2). Cleland (2) has analysed the hexokinase:glucose 6-phosphate dehydrogenase system and presented equations to calculate the transition time, I (see Figure III-1), which take into account the mutarotation of glucose 6-phosphate. Although theoretically sound, the equations are not practical as they do not define the time which must elapse before the primary enzyme rate is approximated by the observed rate. The practical time, tF (t99, see Figure III-1), is not directly obtainable as the 6A equations which accurately define the coupled enzyme systems can not be differentiated. We have thus applied a method for approximating the solutions to these equations which was developed in the previous paper (3). These equations have been verified using hexokinase and phosphoglucomutase as the primary enzymes, and glucose 6-phosphate dehydrogenase as the coupling enzyme. A method of cost minimization is also provided for the system involving 2 coupling enzymes. Materials and Methods Rabbit muscle phosphoglucomutase in 2.5 M (NHA)2SOA’ yeast hexokinase in 3'2.fl (NHA)ZSOA and yeast glucose 6-phosphate dehydrogenase (type IX) were purchased from Sigma Chemical Co. (St. Louis, MO) and used without further purification. The phosphoglucomutase solution was diluted with 5 mM citrate (pH 7.2) prior to use. Addition of the diluted phosphoglucomutase or hexokinase solution did not significantly alter the ionic strength of the assay. All other chemicals were purchased from Sigma and were of the highest quality available. Spectrophotometric measurements were carried out on a Beckman model DU spectrophotometer with a Gilford model 222 photomultiplier at 3A0 nm using a A6 for NADPHI4 of 6.23 X 103lM-1cm-1. All substrate stock solution concentrations were measured using 10 ug of glucose 6-phosphate dehydrogenase (S.A. = 330 I.U.lmg) and 0.66 ug of phosphoglucomutase (S.A. = 210 I.U./mg) or 2 pg hexokinase (S.A. . 321 I.U./mg) at 3A0 nm prior to each set of experiments. A. Abbreviations used: NADP(H); nicotinamide adenine dinucleotide phosphate (reduced form), PGM: phosphoglucomutase, G6PdH: glucose 6- phosphate dehydrogenase Figure III-1: 65 Theoretical time course for a one enzyme coupled assay involving mutarotation of the intermediate as Shown in Scheme 1. The solid line represents accumulation of product and the dashed line is the asymptote to the curve at t . 0. The time taken for [8] to reach 99% of [B] is indicated. The transition time, T, is also Shown. See text for details. 66 [P] 67 Theory One Auxiliary Enzyme A general scheme for a one enzyme coupled reaction in which the intermediate undergoes mutarotation is shown below. Scheme 1 8V1 S > a (I) / k1 k2 V2[B] (1 - a)v / K + [B] 1 > s B > P (I) / (II) \ Q R Where a and 8 denote the concentrations of the two forms of the intermediate. The constant, a, is the fraction of the total velocity of enzyme I which results in the formation of the intermediate a. When enzyme I is hexokinase and S is glucose, a - 0.A (1), because both the a and 8 forms of glucose react with equal facility. The value of a then reflects the equilibrium between a and 8 glucose. If enzyme I is phosphoglucomutase (PGM), a = 1.0 as PGM reacts only with the a form of glucose 1-phosphate. The rate constants for isomerization of the a and 8 forms are represented by the values k and k . For glucose 6-phosphate, 1 2. 1 Cleland (2) gives values of: k = 3.8 min.1 and k - 2.2 min- . v 1 2 1 represents the rate of the initial enzyme (the enzyme under study) and is assumed to be constant over the time course of the experiment. If we assume that the reverse reactions of enzyme I and II are negligible and that V1 is constant, we can define the change in the concentrations of the intermediate over time by equations 1a and 1b. 68 [III-1a] d[a]/dt - av1 + kZEB] - k1[a] [III-1b] d[BJ/dt = (Pan:1 + k1[a] - k2EB] — VZEBJ/(KB + [31) with V defined as 2 [III-2] v2 = v2,max [OJ/(KQ + [0]) where V is the maximal velocity for enzyme II. If K is the 2,max _ B measured Km of enzyme II for a mixture of a and B enantiomers, then (see reference 2): [III-3] K = KB/(l + k 8 /k1) 2 Note that V2 represents the concentration of enzyme II in units minflmL-1 or uM/min. To obtain an exact solution of Equation III-1, we assume that V2 >> v1, and then define [III-A] m = V2/KB Equations 1a and 1b then become: [III-5a] d[a]/dt = av1 + k2[B] — k1[a] [III-5b] d[BJ/dt = (1 - a)v1 + k1[a] - (k2 + m)[8] Integrating equations 5a and b gives [III-6a] [a] = [GJSS + C1exp(r1t) - (EGJSS + C1)exp(r2t) 69 [III-6b] [B] = [BJSS + C2exp(r1t) * ([8]SS + C2)exp(r2t) where: r1 = {-(k1 + k + m) - [(k1 + k + m)2 - Ak1m]1/2}/2 2 2 r2 - {-(k1 + k + m) + [(k1 + k + m)2 - Ak1mJ1/2}/2 2 2 [OJSS = v1(a/k1 + k2/k1m) [BJSS = v1/m C . (av1 + [a]SS r2)/(r1 - r ) 1 2 C2 ~ [(1-a)v1 + [8188 r2J/(r1 - r ) 2 In order to calculate the time required for [B] to achieve a defined fraction of [B]88, we define: FB = [8]/[B]ss. Substituting FB[B]SS for [B] in Equation III-6b gives an expression which relates the time required for [B] to achieve F of [8183 in terms of the enzyme kinetic 8 parameters. [III-7] [8183(F8-1) = c ) 2tF8 26XP0 and k2>O an accurate value (: 0.01%) of t can be obtained from Equation III-8. Fe 1 [111-8] tFB - ;- ln[(1-FB)[BJSS/([B]Ss + 02)] 2 which is derived from Equation III-7 by assuming r'1/r2 > 2 Equation III-7 was obtained by integration of Equation III-1 70 assuming that V >> v . If v is greater than 0.01 V then the error in 2 1 1 assuming that (KB + B) = K 2. 8 becomes significant so that the above assumption is no longer valid. However, an approximate solution can be obtained by rewriting Equation III-1b so that the term (K8 + [8]) is constant. We can then obtain two equations which define an upper (8+) and lower (B_) for the term (KB + [8]). Integration and subsequent numerical analysis of these equations shows that the upper limit yields the best approximation (see Appendix). Solution of this equation for the lag time (tFB) gives Equation III-9. The lag time is denoted by the term F8 t _ as the B+ analysis gives the best estimate. 1 [111-9] tFB_ = ;— 1n[(1-FB)[B]Ss/([BJSS + z 2 )3 1+ where Z1+ = [(1-a)v1 + [BJSerJ/(r1 - r2) and r1 and r2 are defined in Equation III-5 with m = O/K [B]SS is given by Equation III-10: Bf [III-1O] [8188 = v1K8/0 where 0 = V2 - v1. The analysis shown in the Appendix indicates that the tFB- approximation from Equation III-9 can be used for values of vl/V2 up to 0.2. After this point the error is in excess of 10%. Two Auxiliary Enzymes The most common system utilizing two coupling enzymes involves the hexokinase:glucose 6-phosphate dehydrogenase system represented below: 71 Scheme 2 aV2[B] v K + [B] S 1 > B B > a (I) (II) / k1 k2 (1-a)V2[B] V3EBJ K + [B] / K + [B] B > B B > P (II) (III) Q R with a defined as before. We begin by deriving the equations which describe Scheme 2 for the case when KB >> [B] and KB >> [8] (pseudo first order analysis). The following equations use m1 = V2/KB and m2 = V3/KB. [III-11a] d[BJ/dt = v1 - m1[B] [III-11b] d[aJ/dt = am1[B] - k1[a] + kZEB] [III-11c] d[el/dt = (1-a)m1[B] + k1[a] — k2[BJ -m2[BJ Equations 11a, 11b and 110 can be solved for the concentrations of all intermediates: B, a and B (see Appendix). Since we are looking for the time required for the final intermediate to achieve a defined fraction of its steady state value, we present the solution for the intermediate 8 only. [III-12] [8] = [BJSS + D exp(-m1t) + D2exp(31t) + D3exp(32t) 1 Where: [BJSS a v1/m2 RT ‘ k1 + k2 + m2 72 2 1/2 31 - [ RT (RT Ak1m2) ]/2 2 ' 1/2 32 - [ RT + (RT Ak1m2) ]/2 X0 = KCVI/V + av1/k1 X - KB(m2D1 + v1)/¢ D . [k1EBJSS - (1-a)v1]/RT D a ([8]SS + D + 82(XO + X1)/m2)/(32/S1-1) 1 D = -([B]ss + D1 + D2) V = V3 - v1 Note that, as is the case for the one auxiliary enzyme system, if KC is the measured Km of enzyme III for a mixture of a and B enantiomers then KB . KC/(1+k2 [8] and [B], reSpectively, Equation III-12 cannot be integrated. We, /k1). For the case when KB and KB are not much greater than therefore, present equations to approximate t for the two auxiliary FB enzyme case. Following the rationale for the one auxiliary enzyme case, we can obtain two equations which define an upper and lower limit for the concentration of 8. Numerical analysis reveals that the B+ analysis yields the most accurate results and consequently Equation III-13 is the more accurate: [III-13] [8+] = [BJSS + D exp(-¢t/KB) + D 1 2exp(s1t) + D3exp(32t) with D D and 3 obtained from Equation III-12 with m = V/K . 1' 2' D3’ 31 2 2 3 If k a 3.8 min.1 and k = 2.23 min-1, it can be shown that for all 1 2 73 values of V/K > O, s /s 3.2. Equation III-13 then reduces to: B 1 2 [III-1A] [8+] = [BJSS + D1exp(-¢t/KB) + D3exp(32t) The value tFB- is approximated by two relationships under defined conditions. When 32 5 - 0/2KB: [III-15a] tFB- = ln{(1-FB)[B]SS/([8]SS + D1 + D2)}/s2 and when 32 Z -¢/2KB: [III-15b] = -KB ln[(FB-1)[8]Ss/D1]/¢ tFB‘ The conditions defining the application of Equation III-15 Show that the t value depends upon WK F8 or ¢/KB depending upon which coupling enzyme 8 is limiting. As more of enzyme II is added, Equation III-15a predicts the correct tFB- value and when enzyme III is in excess, Equation III-15b predicts the correct t value. If neither of the above conditions F8- applies, then Equation III-1A should be solved numerically to determine the value of tFB- using: [B_] a FB[8]SS and solving for t in a manner similar to that shown by Brooks gt al. (3). A practical limit on the use of equations 1A and 15 must also be imposed. When either v1/V2 or v1/V3 is greater than 0.2, the calculated value of t is in error by about F8 15%. This therefore reflects the limit of the technique. 7A Minimum Concentration of Each Coupling Enzyme For systems involving two auxiliary enzymes as defined above, the transition time, I, (see Figure III-1 and also reference A), is given by [III-16] T = a/k1 + KB/O + KC/Y By solving for V or V 2 3, Equation III-16 gives [III-17a] V - v1 + VKB/(TV - aY/k1 - K ) 2 [III-17b] V3 = v1 + OKC/(re - aO/k C — K ) 1 B In order to minimize the total enzyme concentration, we use the convenient cost function, which is the sum of the price per unit of enzyme multiplied by the concentration of enzyme (see references 2 and 3). or; [III-18] Cost = P2V2 + P3V3 We can substitute Equation III-17 into Equation III-18 and obtain the total cost as a function of either V or V Taking the derivative of 2 3? these functions with respect to V2 or V3 and setting this value to zero gives two equations which are continuous for all values of V and V . The 2 3 ratio of V2 to V gives the slope of the line of minimum cost. 3 75 1/2 V2 P2P3(KB + Tv1 v1a/k1) + P3(P2P3KBKC) [III-19] 6 = - - . . 1/2 V3 P2P3(KC + 1v1 - v1a/k1) + P2(P2P3KBKC) The value of V2 which minimizes the above function is given by the intersection of Equation III-19 and III-17b: R1 + [R12 - AR2(Tk1 - a)]1/2 [III-20] V2 = . 2(Tk1 - a) Where: R1 = v1(1 + O)(k1T - a) + k1(KB + OKC) 2 R2 = v1 6(k11 a) + Ov1k1(KB + KC) Since the I value has no practical use, we use two empirical relationships which represent the average value of the ratio tFB/T: t99 . A.58T and t90 = 2.38 T (see also reference 3). These relationships can be used to obtain an approximate I value for a desired value of tFB' Using the I value, one can obtain minimum values of V2 and V3 which correspond to the I value. One must then return to Equation III-15 and calculate the correct tFB value using the calculated V2 and V3 values. This will ensure that an accurate lag time has been defined. Results To test the theoretical expressions developed in the previous section, a coupled enzyme system involving either PGM or hexokinase as the primary enzyme and glucose 6-phosphate dehydrogenase (G6PdH) as the secondary enzyme was examined. These systems were chosen for their convenience and lack of product inhibition. To apply the theories outlined above, it is necessary to know K the Michaelis constant of B! glucose 6-phosphate for G6PdH. Prior to the present studies, a kinetic Figure III-2: 76 Graph of V9 calculated from Equation III-23 versus the volume of glucose 6-phosphate dehydrogenase added to the reaction mixture. The reaction mixture contained: 10 m! KMOPS (pH 7.2), 10 mi MgCl , 1 mM EDTA, 170 pg NADP , 0.5 mM glucose and 350 ufl ATP. The tEmperature was 30 C and the‘assay was monitored at 3A0 nm. The total volume is 1 mL. For each assay, 0.17 ug of hexokinase was added. The final reaction velocity was 10 nmoleS/min. Comparison of theoretical and actual results for the hexokinase-glucose 6-phosphate dehydrogenase (G6PdH) system. Various amounts of GOPdH were added to 0.17 ug hexokinase and the assay monitored at 3A0 nm. Substrate concentrations were: 170 UM NADP , 500 UM glucose and 350 L: ATP. Other conditions are given in the legend to Figure 111-2. The solid lines are theoretical lines obtained as described in the text. The inset shows the tQQ values for the three cases. V - 10 :_0.5 nmoles/mini’l; 20 UL GOPdH (V = 250 nmole/ too - 1.1 min): 0, 5 uL GOPdH (V - 63 nmole/min, t 1.5 min); 0, 2 UL GCPdH (V2 - 24 nmole/min, Tin) 510 k CS 1 A I ' 09 t99 = 3.5 V2 (n mole/min) [Product] (MM) 200 6 o 20 10 77 5 IO 15 20 pl GGPdH l x y i ' ' 3 ’ T . '99 z - - ' . (MAI, '_ _:_ _ °o 130 260 1 ' (“PC") “I“, 199 o , 1 1 195 0 99 1 1 so - 1 1 , L195 9° 9.0 -2. . . 0.5' 1.0 1.5 ~ 2.0 1 (min) 78 assay was conducted to determine the value of K Under the conditions B7 outlined in the legend of Figure IIIP3, we obtained: KB . 6.1 UM. The value of V2 can also be obtained from this experiment. However, if KB is known and the experimenter wishes only to determine the value of V2 in a single assay, he may proceed as follows. The assay for hexokinase obeys a reaction of the type shown in Scheme 1. At steady state d[a]/dt - 0 and d[BJ/dt - 0 and equations 1a and 1b predict that (see also references 2 and A): [III-21a] [8183 = KBV1/¢ [III-21b] [a]SS = av1/k1 + k2K8v1/k10 The transition time, T, is simply the sum of the steady state concentrations of the intermediates divided by the initial velocity (2,A); and thus equations 21a and 21b give: [III-22] T = a/k1 + KB/¢ which rearranges to give Equation III-23 [III-23] V = v 2 + KB/(T - a/k1) 1 so that a plot of V2 calculated with Equation III-23 using measured values of I (see Figure III-1) versus the amount of enzyme II added (at constant v1) will give the specific activity for the coupling enzyme as shown in Figure III-3. 79 Using the values obtained from Figure III-3, it is possible to construct time courses for each enzyme assay using the value for the observed rate at t - t99 (v1) and the following analysis. The rate of product appearance is given by Equation III-2A: [III-2A] d[PJ/dt - k+[B+] Substituting the value of [8+] from Equation III-28 into Equation III-2A and subsequent integration gives the value of [P] for any time t: [III-25] [P+J = k+{[BJSSt + 21(exp(r1t)-1) - (21+[BJSS)(exp(r2t)-1)} l"1 1‘2 The continuous curves in Figure III-A are constructed using Equation III-25. It is apparent that the theoretically derived time courses (continuous curves) and the observed time courses (symbols, Figure III-A) are in good agreement. Using Equation III-9, we estimate t99 values of: t99 (V2 = 250 uM/min) = 1.1 min, t99 (V2 = 63 uM/min) = 1.3 min and t99 (V2 = 2A uM/min) = 3.A min. The values are in good agreement with the actual t values obtained from computer simulated 99 progress curves using equations 1a and 1b (5). Note that accurate rates are obtained from the experimental progress curves after waiting the required period of time. Figure III-A also shows the t90 and t95 values: the curves appear to be linear after t minutes so that it is hard to 95 95 and t99 minutes. Figure III-5A Shows the results of another experiment using PGM distinguish between the rate at t as the primary enzyme and G6PdH as the coupling enzyme. For this experiment various amounts of PGM were added to a reaction mix containing Figure III-Aa: Figure III-Ab: Figure III-5: 80 Titration of G6PdH with PGM. Various amounts of PGM were added to either 1 unit (0), 2 units (0) or 10 units (A) of G6PdH and the rates measured when the observed velocity was linear. Substrate concentrations were: 30 uM_glucose 1-phosphate, 170 u! NADP and 10 HM glucose 1,6 diphosphate. Other conditions are given in the legend of Figure III-3. Experimental progress curves for measurement of PGM enzyme activity. Top curve, v = 79 nmoleS/min. Bottom curve, v - 3A nmoleS/min. OTHe v b values were obtained from t3 8slope of the dashed lineg.S t and t values (calculated with Equation III-9) are indIEated. 98 - 1 unit. Other conditions are given in the legen8 of Figure III-3. Relationship between t99 and v1/V2 or v1/V3. The values of t99 were obtained using KINFIT A (5) and Equation III-11. Closed symbols represent log(v1/V3) versus t 9 with V . 100 uM/min, K = K - 10 uM. 0, V3 s 50 uM/mig; I,V = 95 uM/min. Open symbols represent log(v1/V ) ver us t with V3 - 100 uM/min, KB - KC - 10 ufl. 0, V2 - 50 uM/mgg; 0,V2 = 95 uM/an. v... (a mole: Imln) 81 A 100 - 75 ' 100 - 50 - (P) (m) 50 - 25 r o 00 0.2 0.4 0.6 no PGM 3 v 2 h . '99 (min) 1 P - -4 _z 0 IOQIV'IV) 82 fixed amounts of G6PdH. Note that, contrary to the expected linear relationship, the observed velocity (yobs) is not linear with respect to the amount of PGM added. The initial velocities plotted in Figure III-5A are the slopes of experimental progress curves (Figure III-SB) when the observed rate appeared to be constant. It appears as if the rate of the primary enzyme decreases with time resulting in decreased Vebs at the longer times. This behaviour was not observed for concentrations of PGM below 0.2 ug/mL. The calculated t95 value for this system, using Equation III-9, is 0.8 minutes. Figure III-5B shows that the reaction velocity depicted by the upper curve is no longer constant at 0.8 minutes and, therefore, an accurate v value cannot be obtained. When v was 1 1 less than A0 nmoleS/min, howeVer, the progress curve depicted by the bottom curve was still linear at t95 minutes. Thus, it is important to calculate t95 values for coupled enzyme reactions to insure that slopes are measured at the proper point in a progress curve. Discussion The equations presented here represent, for the first time, a method for calculating the lag time when the kinetic parameters of the coupling enzyme(s) are known and when one of the intermediates undergoes mutarotation. If one wishes to specify a lag time, the equations can be inverted to obtain the concentration of the coupling enzymes necessary to produce the desired lag period. The development of the equations relies on two assumptions: (a) the rate of the primary enzyme is constant and (b) the reverse reactions are negligible. A third assumption is also implied, i.e., that either v1/V2 or v1/V3 does not exceed 0.2. This 83 latter assumption is necessary as the tF8 estimates are based on approximations of the value V2/(K8 + [8]) or V3/(KB + [8]). Thus the third assunption gives practical limits to the equations. The results obtained with the hexokinase-G6PdH system demonstrate the usefulness of the equations. Using the tFB- values calculated from Equation III-9, accurate primary enzyme rates were obtained from experiments where the coupling enzyme G6PdH was varied (all observed rates were identical) (see Figure III-A). Furthermore, experimentally accurate lag periods (: 10%) were calculated even for the case where v1/V2 - 0.5. A cost minimization technique is described which relies on the minimization of the transition time, T, as defined by Cleland (2). Although the 1 value is not practical, an empirical relationship between T and tFB- is presented. This allows an estimation of a I value from a desired tF8 value. Minimum values of V2 and V3 can then be calculated and used to obtain a more accurate tFB value. It is important to recalculate the tFB value after V2 and V3 have been obtained as the empirical relationships defined are only approximations. Thus the initial tFB may not be the same as the recalculated value. An interesting result was obtained when the exact lag time, calculated with Equation III-11, was plotted against log(v1/V3) or log(v1/V2). Figure III-2 shows that the t99 values differ depending on whether V is greater than V 2 > V2 (and 3 or the reverse is true. When V3 K8 . KB). the lag time is shorter. This is because the system in scheme 2 is not symmetrical about the intermediate 8. It can also be shown that a does not achieve 0.9 or 0.99 of its steady state concentration when [BI/[BJSS = 0.9 or 0.99 and a=0.A, even though the observed velocity is equal or greater than 0.9 or 0.99 times the initial velocity. When a = 8A 1.0, the intermeditate a does achieve the indicated FB value because it is an obligatory intermediate in reaction Scheme 2. Finally it should be noted that, when the intermediates undergo mutarotation, the observed lag time (tFB) increases condiderably compared to the case when intermediates do not mutarotate. The tF values in Figure III-5B were obtained by assuming that, with the PGM-G6PdH system, mutarotation of one of its intermediates did occur. If the calculations were made assuming that mutarotation did not occur as outlined in the preceeding paper (3). we obtained a t value of 0.06-0.08 minutes 95 (depending on v1). Thus one might assume that the linear portion of the progress curves accurately reflected the primary enzyme velocity. However, the proper calculation shows that t = 0.8 minutes. At this 95 time, the observed rate is obviously not linear and consequently other factors (such as decreasing enzyme activity) may be contributing to the observed velocity. This underscores the importance of calculating tF values derived from a model which accurately reflects the reaction scheme. References 1. Wurster,B. and Hess, B. (1973). Eur. J. Biochem., 36, 60-75. 2. Cleland, W.W. (1979), Anal. Biochem., 29, 1A2-1A5. 3. Brooks, S.P.J., Espinola, T. and Suelter, C.S., Can. J. Biochem. Cell Biol., 62, 9A5~955. A. Easterby, T.S. (1981) Biochem. J., 122, 155-161. 5. Dye, J.L. and Nicely, V.A. (1971), J. Chem. Educ., £8, AA3-AA8. 85 Appendi x One Auxiliary Enzyme In order to obtain an approximate equation for the lag time, we rewrite Equation III-1b so that the term V2/(KB + [8]) is constant. Equations 1a and 1b then yield an explicit value for tF Therefore, by 87 choosing constant values of VZ/(K + [8]) which approximate the true 8 value of 8 at any time t, we can rewrite Equation III-1b as: [III-A1a] d[B+]/dt (1-a)v1 + k1[a] - [8](k2 + k+) [III-A1b] d[B_]ldt (1-a)v1 + k1[a] - [8](k2 + k_) where 8+ and 8. reflect the largest and smallest possible values of B. The term k+ is now defined as the smallest value of V2/(K8 + [8]) such that d[B+]/dt is always greater than d[BJ/dt or: k+ = VZ/(KB + [BJSS) = o/KB where 0 - V2 - v1. Conversely k_ is the largest value of V2/(KB + [8]) or k_ - V2/KB. Using these approximations we now integrate equations 26a, 26b and 1a. This integration gives two equations of the general form: [III-A2] [B] = v1/k + Z exp(r1t) + Z exp(r2t) 1 2 "1th r1 and r2 as defined in Equation III-5 and with m = k, or k-. Z1 and 22 are obtained by selecting an appropriate initial (t=0) condition. 86 Selecting [8+]t=0= 0 gives: [III-A3] [8+] = [B]SS + Z1+exp(r1t) - 22+exp(r2t) where Z = [(1-a)v1 + [BJSSPZJ/(r1-r2)’ Z 1+ = -([B]ss+z1+) and r and r 2+ 1 2 are defined by Equation III-5 with m = ¢/KB. [M83 is given by Equation III-10. If we choose [B-]t=O= 0, Equation III-A2 gives Equation III-5 once again. However, a better estimate of tFB can be obtained if we let k - k_ and Z = Z in Equation III—A2 (i.e. [8_] 2 2+ < 0). Equation III-A2 t=O then gives [III-AA] [B_] = V1KB/V2 + Z1_exp(r1t) - ([8]SS + 21+)exp(r2t) where Z1_ . [(1-a)v1 + v1KBr2/V2]/(r1-r2) and r1 and r2 are obtained from Equation III-5 with m = V2/KB. Figure III-A1a shows a comparison of the time courses for [B]/KB exact (Equation III-1), [B_J/KB (Equation III-AA) and [6+J/K8 (Equation III-A3). The curves are drawn for V . 100 uM/min and v - 10 uM/min. 2 1 Note that both [SJ/K exact (closed triangles) and [8+J/K 8 (open circles) B tend toward [BJSS/K when t -> m, the value of [B_J/KB (closed circles) 8 tends toward v1KB/V2. If v1/V2 decreases, all curves tend toward [BJSS/K because v K /c tends toward v K /V as v /V decreases. This is B 1 8 1 8 2 1 2 clearly shown in Figure III-A1b where the error in t is plotted as a F8 Figure III-A1a: Figure III-A1b: 87 Theoretical time courses for the accumulation of [8] calculated from three sources. 0, Equation III—AA (8_); 0, Equation III-A3 (8+) and A, [B]/K8 exact from KINFIT A estimation (5). Relationship between the error obtained from the tF analysis and the true tF8 value as a function of v17V2. Solid line: K /V = 1; dashed line, K /V2 - 0.1, dotted- dashed line, A / 2 = 0.01; double-dotged dashed line; first order apgroximation (see Equation III-6). The value of ERR is given by: ERR - [tFB(true) - tFB(estimate)]/tF8(true). 88 A - /£p]ts/Kp 0.06 ”AAA—oaa—a—o—F—p-.. [PI/KP 0.5 ' 0.04 ERR 0.25 0.02 o 0 L L Q '0 2.0 - 0.01 89 function of v1/V2 for various values of K /V2 as indicated in the figure 8 legend. As KB/VZ decreases, the error in the determination of tF8 decreases (compare solid line, dashed line and dotted-dashed line). The estimate from Equation III-6, which assumes that V >> v 2 1 (m = V2/KB, double-dotted-dashed line, Figure III-A1b), has an error considerably greater than that for the t estimate (dashed line). F8- The values of tFB- and tF8+ are also obtained from equations A3 and AA by defining F as before with [BJSS given by Equation III-10 when 8 [III-A5a] tFB_ = 1n[(1-F8)[8]SS/([BJSS + 21+)J/r2 [III-A5b] tF8+ - 1n[(¢/V2 -FB)[8]ss/([8]Ss + 21+)]/r2 Two Auxiliary Enzymes Solving Equation III-11 for all concentrations of intermediates gives: [III-A6a] [B] = [B]SS - [BJSSexp(-m1t) [III-A6b] [a] - [OJSS + C1exp(-m1t) + C2exp(s1t) + C3exp(52t) and Equation III-12. By defining boundary conditions of [B]t=0 = [a] = [B]t=0 = 0, we obtain: [B]SS = v /m1 1 [GJSS = av /k + k v1/k1m 1 1 2 2 C a 1 (3V1 + [BJSSm2 + [Bjssk2)/RT C2 = (v1 + s,[8]SS + D,(m2+31) + (XO+X1)(k1+52))/(s1-sz) t=O' 90 C3 - -([a]ss + C1 + C ) 2 where m1 = V /KB, m = V /K and [B]Ss, D X 2 2 3 B by Equation III-12. X1, S and S as defined 1’ 0’ 1 2 Following the rationale for the one auxiliary enzyme case, we choose approximations for the term V /(KB + [8]) which will allow 3 integration of the resulting differential equations. Thus, as before, we choose r+ and r_ values which represent the highest and lowest value for the expression V3/(KB + [8]): where V a V3 - v1. Using the k+ and k_ values from the one coupling enzyme case, we can obtain two equations which represent an upper and lower limit for the concentration of 8 over the time course of the experiment. Computer simulation (5) indicated that Equation III-13 is the more accurate and it is, therefore, presented in the text. Chapter IV Characterization of Chicken Atrium Mitochondrial Creatine Kinase Purified Using Transition State Analog Affinity Chromatography Abstract A method for preparing homogeneous mitochondrial creatine kinase from chicken ventricle is presented. The two column procedure, which can be completed in two days, uses Procion Red-agarose and Agarose-Hexane-ADP column chromatography. The latter column is run under conditions which promote the formation of a transition-state analog. The enzyme is a dimer composed of two A3,000 molecular weight subunits. The sequence of the first N-terminal 20 amino acids shows that the enzyme is different from the cytosolic isozymes but similar to human mitochondrial creatine kinase. The enzyme has an extinction coefficient of e - 2.1 + 0.A 280 '— mL-mg-1ocm 1 and a specific activity of 12A IU/mL. The kinetic constants for the chicken heart mitochondrial isozyme are comparable to values for the canine and beef heart enzymes. Introduction Creatine kinase (EC 2.7.3.2) exists in nature in several isozymic forms (1). Two cytosolic subunits, M (muscle)3 and 8 (brain), dimerize to form three different cytOplasmic isozymes: MM, BB and the hybrid MB (2). The MM isozyme of creatine kinase is found in mature skeletal muscle and mammalian myocardium, the BB isozyme in mammalian brain, 91 92 Ileaural tissue and embryonic skeletal muscle and avian myocardium (1); and ‘tiue hybrid MB creatine kinase appears in mammalian heart and skeletal tntnscle (1). An additional CK isozyme, positively charged at pH 8.8, was iJTitially reported in rat heart and brain mitochondria (3). Later :studies revealed that human, beef, and rat heart, as well as rat brain, sskeletal muscle and intestinal muscle mitochondria contain significant amounts of the mitochondrial isozyme of creatine kinase (MiMi-CK). Nominal amounts of MiMi-CK appear in rat and rabbit liver, kidney and testes (A). Although initial reports indicated that chicken heart mitochondria did not contain MiMi-CK (5), subsequent studies have confirmed the presence of the isozyme in small amounts (6). MiMi-CK is associated with the outer surface of the inner mitochondrial membrane in all tissues containing the isozyme (A, 7-10). Although all CK isozymes are dimeric and have similar kinetic constants (11, 12), the MiMi-CK subunits are different from the cytoplasmic CK subunits as shown by amino acid composition analysis (12-1A), N-terminal sequence analysis (15, 16). lack of antisera cross reactivity (13, 1A, 17, 18) and the absence of hybridization with the cytOplasmic subunits to form hetero-dimeric enzyme (13, 1A, 17, 18). Previously published purification procedures for MiMi-CK take advantage of an ionic-strength-dependent release of this enzyme from the inner mitochondrial membrane (12-1A, 17-20). Incubating intact 3. Abbreviations used: B; brain type creatine kinase, BICINE: N,N-bis(2- hydroxyethyl)g1ycine, BSA; bovine serum albumin, CK; creatine kinase, Hepes: N-2-Hydroxyethylpiperazine-N'-2-ethanesulfonic acid, IU; 1 umole of substrate converted per minute, M; muscle type creatine kinase, MiMi- CK; mitochondrial creatine kinase, MOPS; 3-(N-morpholino)propanesulfonic acid, NaDOC: sodium deoxycholate, PMSF: phenylmethylsulfonyl fluoride, TPCK: N-tosyl-L-phenylalanine chloromethyl ketone. 93 Iniqtochondria in 100 mg sodium phosphate releases the enzyme (20) which is :stnbsequently purified by anion exchange, gel filtration, and ATP-affinity cfluromatography (12-1A, 17-20). Several of these procedures are rather lengthy and incorporate unnecessary steps. Varying specific activities (12; 1A, 17) indicate that proteolytic products or enzymes with altered cactivity may be present in the final enzyme preparation, even though a single band is obtained after SDS-polyacrlyamide gel electrophoresis. This paper presents a procedure for the purification of homogeneous chicken ventricle MiMi-CK using dye-ligand affinity and transition state analog chromatography. This two column procedure can be completed in approximately two days and gives a high yield as compared to previous procedures. The amino acid content, 20 N-terminal amino acid sequence, extinction coefficient, kinetic parameters, molecular weight and subunit composition are reported. Materials and Methods Materials: All chemicals, enzymes and creatine kinase assay kits were obtained from Sigma Chemical Co. (St. Louis MO) unless otherwise specified. Agarose-Hexane-adenosine 5'diphosphate, type 3 (Agarose- Hexane-ADP) was obtained from P.L. Biochemicals, Inc. (Milwaukee, WI). Sephraphore III cellulose acetate electrophoresis strips were purchased from Gelman Sciences, Inc. (Ann Arbor, MI). Common laboratory chemicals were reagent grade or better. NaDOC was prepared from deoxycholic acid recrystallized from hot 80% acetone. Chickens were obtained from the Department of Animal Science, Michigan State University. 9A Enzyme Assays and Protein Determinations: GK activity was determined spectrophotometrically at 3A0 nm at 30°C using the CK assay mix from Sigma (21, 22). The concentration of cytochrome 233 was determined from the differences in absorbance of the reduced minus oxidized spectra at 1cm_1, 23). Protein concentrations were 602 minus 630 nm (Ac - 211 mM- determined by fluorescamine assays (2A) using bovine serum albumin as a standard. Cellulose Acetate Electrophoresis: CK isozymes were separated on 2.5 X 17 cm Gelman Sephraphore III cellulose polyacetate electrophoresis strips in 0.06 M Tris-barbital, pH 8.8, 25 mM 2-mercaptoethanol as previously described (20). The electrophoresis buffer contained 1% Triton X-100 to prevent CK from sticking to the strips. Electrophoresis proceeded for two hours at 300 V at 6°C. Following electrophoresis, the strips were stained for CK activity as previously described (25). Polyacrylamide Gel Electrophoresis: Homogeneity and subunit molecular weights of crude and purified MiMi-CK were determined using sodium dodecyl sulfate-polyacrylamide gel (10% polyacrylamide, 0.26% bisacrylamide) electrophoresis (SDS-PAGE) in the presence of 2- mercaptoethanol according to the method of Laemmli (26). Molecular weight standards were a mixture of bovine serum albumin (68,000), chicken egg albumin (A3,000), glyceraldehyde 3-phosphate dehydrogenase (36,000) and B-lactoglobulin (18,000). After electrophoresis, the gels were fixed for 1 hour in 10% acetic acid, stained for 2 hours in 0.25% Coomassie brillant blue R dissolved in 50% methanol plus 7.5% acetic acid and destained in 25% methanol plus 10% acetic acid. 95 Sequencing and Amino Acid Analysis: The sequence of the first 6 N- terminal amino acids and the total amino acid composition of MiMi-CK were provided by the Michigan State University Macromolecular Structure Facility. The first 20 N-terminal amino acid sequence was graciously provided by Dr. A. W. Strauss (Washington University School of Medicine, St. Louis, Mo). The amino acid composition was determined as follows. A sample of protein was hydrolyzed under nitrogen in the presence of 6‘5 constant boiling HCl and phenol at 1100C for 2A hours. The sample was then neutralized by adding 10 uL of a 2:2:1 mixture (by volume) of ethanol:water:triethylamine, dried under vacuum and the neutralization procedure repeated once. The amino acid composition was measured, after derivitization with phenyisothiocyanate, on a PICOoTAG (Waters Instruments) column and the concentration obtained by peak integration. The results are reported as the mean of two runs. In order to prevent cysteine oxidation, the protein was carboxymethylated with iodoacetic acid according to Gracey (27) prior to amino acid composition analysis. No attempt was made to analyze for tryptophan. Equilibrium Centrifugation: The molecular weight of native MiMi-CK was determined with a Beckman airfuge as previously described (28, 29). Including tritiated water in the protein solution made it possible to accurately determine the volume of each succeeding sample in the equilibrium gradient as suggested by Nickerson at al. (30). 96 Carboxypeptidase Y digestions: Time dependent digestions of MiMi-CK 'with carboxypeptidase Y were performed by incubating 75 ug of carboxypeptidase Y with 170 uL of 0.53 ug/mL MiMi—CK in 10 mM MES, 2% glycerol, 25 mg 2-mercaptoethanol (pH 6.0) and and 20 UM pepstatin A (31) at room temperature. The reaction was stopped at the indicated time points by adding PMSF to a final concentration of 2 mM. Determining Kinetic Constants: Kinetic parameters for MiMi-CK were determined using coupled enzyme reactions at 300C. The forward reaction (creatine phosphate production) was measured using excess pyruvate kinase and lactate dehydrogenase in a buffer containing 10 mM MOPS, 2.5 mM magnesium acetate, 0.1 mM_EDTA, 50 mM potassium acetate, 2.5 mM dithiothreitol, 2.5% glycerol, 0.A8 m! NADH, and 1.8 mM phospho(enol)pyruvate at pH 7.0. The reverse reaction (creatine production) was measured using hexokinase and glucose 6-phosphate dehydrogenase in a buffer containing 25 mM MOPS, 75 mM sucrose, 225 mg mannitol, 3.5 mM magnesium acetate, 1 mM EDTA, and 2 mM dithiothreitol at pH 7.A. The amount of coupling enzymes needed to perform these assays was determined using Equations II-22 and II-30 (Chapter II) for pyruvate kinase and lactate dehydrogenase, and Equations III-1A and III-20 (Chapter III) for hexokinase and glucose 6-phosphate dehydrogenase. The MgATP-2 and MgADP- concentrations were calculated using the equilibrium constants for the formation of these species (32). Purification Procedure All purification steps were performed at room temperature unless otherwise noted. Glycerol or Triton X-100 was used throughout this study 97 to prevent loss of enzyme by adsorption to the walls of vessels or chromatographic matrices (35). The pH of all buffers was measured at room temperature. Isolating Mitochondria and Preparing Mitoplasts: Mitochondria were isolated from eight 6-8 week old white leghorn chickens in ice cold 75 mM sucrose, 222 mM mannitol, 5 mM EDTA, 25 mM 2-mercaptoethanol, 2 mM PMSF, 0.1 m! TPCK (Buffer A). Extracting the mitochondria twice increased the yield (23). MitOplasts were prepared from mitochondria using digitonin (Chapter IV). Releasing MiMi-CK from Mitoplasts: After washing mitoplasts with Buffer A, the pellet was resuspended in 15 mL of 50 mM sodium phosphate, 10 nM Tris (pH 8.0) in Buffer A and allowed to incubate at 30°C for 7 minutes (see Chapter V). The mitoplasts were centrifuged at 8,000 x g for 10 minutes and the supernatant was saved. This procedure was repeated and the pooled supernatants were added to 70 mL of 25 mM 2- mercaptoethanol, 1.A3% NaDOC, 0.36% (w/v) Triton X-100 and the pH adjusted to 8.2. This solution has a Specific activity of 1.5 IU/mg (Table IV-1). Procion Red-agarose Chromatography: The 5 mL Procion Red-agarose column was washed with 50 ml of 1% (w/v) sodium dodecyl sulfate and subsequently with 50 mL each of solutions made up of 30:5:5:80, 85:5:5:5 and 20:0:0:80 of acetone:triethylamine:acetic acid:water (33). The column was then washed with 50 mL water and equilibrated with 20 mM Tris- HCl (pH 8.2), 1 mM EDTA, 25 mM 2-mercaptoethanol, 0.25% (w/v) Triton X- 98 100, 2 mM PMSF and 0.1 m! TPCK (Buffer B) plus 1% (w/v) NaDOC. The sample from step b was loaded directly onto the Procion Red-agarose column and washed with 50 mL 1% NaDOC in Buffer B to remove contaminating BB-CK and other non-binding proteins. MiMi-CK is then eluted from the column with 2.5 M NaCl in Buffer B. Occasionally the first fraction contains a red pigment which persists during subsequent purification steps if not removed. For this reason, it is useful to collect 3 or A small fractions (1-2 mL each) at the beginning of the elution and discard those with a red pigment. The fractions containing activity (normally the first A0 mL) are pooled and dialyzed overnight against two changes (750 mL each) of 20 mm Hepes (pH 7.A), 1 mM EDTA, 25 mM 2- mercaptoethanol, and 0.25% (w/v) Triton X-100 (Buffer C) at 600 to remove the NaCl. The pool of GK activity from the Procion Red-agarose column contains only the MiMi‘CK isozyme (data not shown) and has a specific activity of 11 IU/mg (Table IV-1). Affinity Chromatography: The 2 mL Agarose-Hexane-ADP column was washed in the same manner as the Procion Red-agarose column and equilibrated with 50 ml of Buffer 0. Following dialysis of the sample from the Procion Red-agarose column, the pool of MiMi-CK activity was applied to the Agarose-Hexane-ADP column and the column was washed with 50 mL of Buffer C to remove non-bound proteins. Washing with 50 mL Buffer C containing 5% glycerol instead of Triton X-100 removes Triton X-100 from the final sample. The disappearance of Triton X-100 can be followed by monitoring the drop size until it appears constant (change of surface tension). The column is then washed with 50 mL of 20 mM BICINE (pH 8.0), 25 mM 2- mercaptoethanol, 2% (v/v) glycerol, 20 mg creatine, 30 mg KNO 2 mM 3’ 99 Table IV-1: Purification summary for the preparation of MiMi-CK. Purification Volume [Protein] Total Specifica Yield Step activity activity (mL) (mg/mL) (IU) (IU/mg) (%) Crude solution 200 11.9 11,200 --- 100b Mitochondria 5 29.7 1A6 1 33 Released enzyme 100 0.36 53 2 12 Post Procion Red-agarose A0 0.11 A6 11C 10.3 Final enzyme 3.1 0.10 3A 350 7.5 Note: a. The specific activity is based on the fluorescamine procedure using BSA as a standard . b. 100 percent yield is based on the observation that A% of the total ventricular CK activity is due to the mitochondrial enzyme (data not shown). 0. Determining the protein concentration by several spectrophotometric methods gives a specific activity of 12A IU/mg (see Table IV-2). 100 Figure IV-1: ADP (Transition state analog) column profile. The dialyzed peak fractions from the Procion Red-agarose step were loaded on a 2 mL Hexane Agarose-ADP column in Buffer C. The column was washed with Buffer C and Buffer C containing 5% (v/v) glycerol instead of TX-100. Washing with Buffer D (arrow) Shows a large release of protein (0) and a small release of enzyme activity (0). After the activity has decreased to zero, the enzyme is eluted with a 20 mL gradient of 0 - 0.6 M NaCl. The NaCl concentration (0) is estimated from the conductivity of the fractions. Fractions at a volume of 156.8 to 167 mL were pooled to give the final purified enzyme. 101 3.5 .529. on. 00. on o o . o .... o o. r 0N . mud N on .. 3 323: 6. cos 36x03 8 .. v6 522; e on ad. ad - e _ _ _1 E as gosz“. Figure IV-2: 102 SDS-Polyacrylamide Gel Electrophoresis of Samples at various stages of purification. Samples were loaded on a 10% polyacrylamide gel according to the procedure outlined in the Methods section. Lanes 1,2 7 and 8 are standards with molecular weights indicated on the Figure. Other lanes are as follows: Lane 3, enzyme released from'the mitoplasts; Lane A, dialyzed post Procion Red-agarose peak fractions: Lane 5, proteins released from Hexane Agarose-ADP column after washing with Buffer D; Lane 6, purified MiMi-CK. 8 6 5 4 3 133 D b c . I ' 0 " ‘ ”"’ I I I Q i "0 '..«p 1 ]I 1 I '° 1 -..M .. * l'. —‘- . . . 0 10A MgCl2 (Buffer D, see reference 36). This wash releases approximately 25% of the bound enzyme (Table IV-1). The CK activity in the eluant should be zero before the NaCl gradient is applied (Figure IV-1). Purified MiMi-CK is eluted in a single peak from the Agarose-Hexane-ADP column with a 20 mL gradient of 0-0.6 M NaCl in Buffer D. Approximately 1 mL fractions are collected. Contaminating protein, when present, is located in the first few fractions of the gradient and consequently these fractions are monitored closely for Specific activity. Concentrating and Storing the Enzyme: The pooled fractions from the ADP column are placed in a dialysis bag which is surrounded by solid sucrose suspended on a piece of cheese cloth in a covered chamber. The solution is allowed to concentrate for 2-3 hours and is then dialyzed against 1 liter of 10 mM MOPS (pH 7.2), 2% (v/v) glycerol, 25 mM 2- mercaptoethanol, and 0.1 mM EDTA overnight at 60C. The enzyme can be stored at A00 for longer than A months without noticeable loss of kinetic or binding activity provided the 2-mercaptoethanol concentration is kept high (sealed tubes with minimal opening and closing). This enzyme has a specific activity of 350 IU/mg when the protein is measured by fluorescamine (Table IV-1) using BSA as a standard. Results Procion Red-Agarose Chromatography: Chromatographing MiMi-CK on a Procion Red-agarose column in the presence of Triton X-100 requires the addition of 1% (w/v) NaDOC because MiMi-CK fails to bind to Procion Red- agarose in 0.25% (w/v) Triton X-100. The NaDOC presumably forms mixed charged micelles with Triton X-100 which excludes the dye from the 105 micelle (37). The Procion Red dye is now free to react with MiMi-CK: BB- CK does not bind to the column. Procion Red-agarose chromatography, like cibacron Blue-agarose chromatography (35) affords a convenient method for separating BB-CK from MiMi-CK. Note that the elution buffer contains NaCl but not NaDOC which is not required for the elution as it functions to enable MiMi-CK binding to the dye. Transition State Analog Affinity7Chromatography: Using an Agarose- Hexane ATP affinity column, as originally used by Hall 32 al. (12, 20) to purify bovine heart MiMi-CK, failed to yield homogeneous enzyme. However, using an Agarose-Hexane-ADP column and conditions which promote the formation of a transition state analog yielded homogeneous enzyme. The initial wash of the column with Buffer D results in a 25% loss of the enzyme activity (Figure IV-1, Table IV-1). This loss is not due to overloading of the column because the dialyzed enzyme fails to bind to a clean Agarose-Hexane-ADP column equilibrated with Buffer D. This non- binding enzyme was not studied further. Combining fractions with a Specific activity greater than 200 IU/mg gives a preparation which shows a single band on an SDS-polyacrylamide gel (Figure IV-2). Purity and Extinction Coefficient: Three independent criteria Show that the purified enzyme is homogeneous. A single band is present after SDS-polyacrylamide electrophoresis of 15 ug of enzyme (lane 6, Figure IV-2). The specific activity of the peak fractions from the ADP column were constant (Figure IV-1). The first cycle of the amino acid sequencing yielded a single N-terminal amino acid >98% pure. Determining the specific activity of MiMi-CK depends on the choice 106 Table IV-2: Measuring MiMi-CK protein by Spectrophotometric methods. Procedure (reference) Extinciton Protein concentration coefficient BSA MiMi-CK --------- (mL-mg-1-cm-1) (WE;;L) (8878;; Fluorometric ----- 80.0a 10.11 1 0.2 Spectrophotometric: 5 (38) 32.3 87.2 31.5 A215-A225 (39) 6.9A 81.6 29.5 A22A-A233 (A0) A.7A 70.5 28.3 0 (A1) 2.113b 80.1 28.8 A 80 “m 2:520 7“ _____ 3.7.1..-- Mean spectrophotometric: 79.8 I 6.0 29.2 1 1.5 Note: MiMi-CK was dialyzed against 50 mM sodium phosphate (pH 7.A) containing 5% (v/v) glycerol for A hours to remove 2-mercaptoethanol. The BSA solution was prepared in 50 mg sodium phosphate (pH 7.A) containing 5% (v/v) glycerol. a. BSA at 80 ug/mL was used aS'a basis for both the fluorescamine and spectrophotometric methods. 0 and c. The extinction coefficients for these cases were calculated using b: 6280 = 3A.1A8*(A280/ A205)- 0. O2, and c: €280 - 30.0*(A280/ A207) -0. 05. 107 Table IV-3: Amino acid composition of MiMi-CK. Amino Acid % of totala Amino Acid % of total3 Asp + Asn 13.1 Tyr 0.A Glu + Gln 9.7 Val A.7 Ser 6.0 Met 1.6 Gly 19.2 Cys 1.8 H18 '2.5 119 1.6 Arg 7.2 Leu 6.0 Thr A.8 Phe 1.1 Ala 9.A Lys 2.8 Pro 8 1 ‘ Note: a. The amino acid composition is presented as a percentage of the total ananlyzed (tryptophan is not included). Figure IV-3: 108 N-Terminal sequences for creatine kinase from chicken tissues. Chicken muscle (M) and brain (B) amino sequences are taken from reference A3. Cytoplasmic CK amino acids which are identical to the mitochondrial isozyme appear in bold type. See Methods section for experimental details. The 11 Nsterminal amino acids from human heart MiMi-CK sequence are taken from reference 17. Isozyme: M: B: Mi: Mi: Mi: 109 1 5 10 15 Pro-Phe-Ser-Ser-Thr-His-Asn-Lys-His-Lys-Leu-Lys-Phe-Ser-Ala- Pro-Phe-Ser-Asn-Ser-His-Asn-Leu-Leu-Lys-Met-Lys-Tyr-Ser-Val- (chicken) Thr-Val-His-Glu-Lys-Arg-Lys-Leu-Phe-Pro-Pro- (human) Glu-Val-Cys-Glu-Cys-Thr-Ser-Leu-Phe-Pro-Pro 1 5 1O 16 20 25 30 Glu-Glu-Glu-Phe-Pro-ASp-Leu-Ser-Lys-His-Asn-Asn-(His)-Met-Ala : Asp-Asp-Glu-Tyr-PrOPAsp-Leu-Ser- ? -His-Asn-Asn-(His)-Met-Ala Ser-Ala-Asp-Tyr-Pro-Asp-Leu-Arg-Lys 15 2O 110 of both the method used to measure the protein concentration and the type of protein used for the standard curve (39 - A1). In order to minimize the problem of varied responses of proteins, we chose to measure the protein concentration using spectrophotometric procedures which are largely independent of the type of protein standard. The results of these measurements are presented in Table IV-2. Protein determinations using the fluorescamine method are included for comparison. Note that the response of MiMi-CK to fluorescamine is very different from BSA; fluorescamine underestimates the protein concentration by approximately 2.8 fold. The spectrophotometric methods give a specific activity for the pure enzyme of 12A IU/mg using the Sigma creatine kinase assay kit in the direction of creatine synthesis at pH 6.9. The extinction coefficients for the purified protein are c . 2.1 I 0.A mL-mg-1-cm 1, -1 -1 and £205 a 31.7 :_2.5 mL mg cm _. 280 Amino Acid Composition and Sequences: The amino acid composition of purified MiMi-CK, in percentages of the total amino acids present in the analysis, is presented in Table IV-3. Note that both glutamine and asparagine have been hydrolyzed to the carboxylic acids and are included with glutamic acid and aspartic acid, respectively. The N-terminal sequence of the homogeneous enzyme is presented in Figure IV-3. For comparison the N-terminal sequences of the M and B isozymes from chicken and from human mitochondria are also presented. Although extensive homology exists between the two cytosolic subunits in the first 30 amino acids (19 amino acids are identical), the 20 amino acids from the N terminal of the chicken heart mitochondrial enzyme are clearly different from the cytosolic enzymes. The seven cytosolic CK 111 amino acids which are identical to the mitochondrial isozyme are indicated in bold type. Note that the MiMi-CK sequence has been shifted by 5 amino acids so that the greatest homology exists between the three sequences. When the 10 N-terminal amino acids for the two mitochondrial CK enzymes are compared to one another, 6 out of ten are identical (Shown by underlining). C-Terminal Studies: Figure IV-A Shows the results of experiments to define a binding domain for MiMi-CK. Treating intact protein with carboxypeptidase Y in the presence of an endopeptidase inhibitor pepstatein A (31) results in a rapid decrease in the catalytic activity but little, if any, alteration in the ability of MiMi-CK to bind to mitoplasts. These results agree with previous studies which localized active site essential amino acid(s) for the cytoplasmic M form of the enzyme near the C-terminus (A2, A3). Kinetic Studies: Kinetic analysis of the purified enzyme gives results which are consistent with a rapid equilibrium random ordered mechanism identical to other creatine kinase isozymes. The kinetic constants for the various processes are presented in Table IV-A. The Ka value for creatine phosphate measured in our system is about 10 times higher than that reported by Hall gt al. (13) but is close to the values reported for other CK enzymes (11). The Km values for MgATP.2 are approximately five fold lower than the cytosolic enzyme values in agreement with previous values for rat heart MiMi-CK (A). Figure IV-A: Figure IV-5: 112 Kinetic activity and binding ability of carboxypeptidase Y treated MiMi-CK. After treatment of MiMi-CK with carboxypeptidase Y, approximately 0.7 IU of enzyme was added to mitoplasts (0.A3 nmole cytochrome as , prepared according to 36) in 0.1 mL of 75 mM sucrose, 25 mM mannitol, 0.2% BSA, 2 mM dithiothreitol and 10 mM MES (pH 6.A). The tubes were incubated at 300C for 10 minutes, centrifuged at 8,000 x g for 10 minutes and the supernatants removed and assayed for enzyme activity. The percent bound represents the supernatant activity divided by the total activity present. The control tubes are represented by the solid symbols and the protease treated samples are represented by the open symbols. Determining the molecular weight of MiMi-CK by equilibrium sedimentation. MiMi-CK (0.16 IU) was placed in 150 uL of 12 mM HEPES, 25 mM 2-me§captoethanol, 1 mMEDTA, 0.2 (w/v) dextran and 0.1 uCi of H 0 (pH 7.A). The sample was centrifu ed for 25.5 hours at 29.85 K rpm in a Beckman Airfuge at 6 C. Samples were carefully removed, the radioactivity counted to determine their volume and assayed for CK activity. 113 ~~ ' 0.4 CKoct (1U/mL) (2,13) - 0.2 100 ~ 70 000110 __o_____T (0,0) A\ \ 50 — \ \ \ \3\\ \ \ \\A._ o l 0 90 0 - "‘(CKOCL) -| r -2 - 1.0 11A Table IV-A: Kinetic constants for MiMi-CK. Constant Value (mM) Forward reaction (pH 7.0): Ka (MgATP—2) 0.25 1 0.03 Ka (MgATP-2) 0.13 1 0.02 Kb (creatine) 20 1_3 Kb (creatine) 10 1 1 VI. 200 1 20 IU/mg Reverse reaction (pH 7.A): KC (creatine phosphate) 6‘1 2 Kc (creatine phosphate) 2 1 0.A Kd (MgADP') 0.08 1 0.02 Kd (MgADP-) 0.03 1 0.01 Vr 210 1 3A IU/mg Note. The constants are defined in Equation I-2. Reaction conditions are described in the Materials and Methods section. For the forward reaction, the creatine concentrations were 50 mM, 35 mM, 20 mM, 10 mM, 7 mM, 5 mM, and 3 mM and the MgATP concentrations were 790 uM, 561 uM, -370 uM, 197 uM, 98 uM, 56 HM, and 35 UM. For the reverse reaction creatine phosphate concentrations were 28 mM, 21 mM, 17.5 mM, 1A mM 12. 6 mM, 10. 5 mM, 7.0 mM, 5. 6 mM, 3. 5 mM, 2.1 mM, and 0. 7 mM, MgADP concentrations were 212 uM,1B2 uM,109 uM, 55 uM, 22" uM, and 11 (TM The magnesium nucleotide concentrations were calculated as described by Storer and Cornish-Bowden (32). The data were analyzed using a computer program (see Appendix A) and the results are presented as': one standard deviation. 115 Molecular Weight Studies: A molecular weight of 8A,000 :_5,000 was determined by equilibrium centrifugation according to Pollet 33 a; (29). This molecular weight agrees well with the subunit molecular weight estimated from its mobility relative to a standard protein mixture on a 10% acrylamide gel (Figure IV-2). Like other CK isozymes, the enzyme is a dimer consisting of subunits of A3,000 molecular weight. Discussion The protease inhibitors PMSF and TPCK are included in the initial stages of the preparation to prevent partial degradation of the protein which may give erroneous amino acid composition and N-terminal sequence results. These two inhibitors are used because of previous studies which indicated that proteases sensitive to these inhibitors are present in mitochondria (A5, A6). Although Procion Red-Agarose column chromatography increases the specific activity by only a small amount, this step has been retained for two reasons: a) it separates the cytosolic isozyme from the mitochondrial isozyme (33) and b) it removes proteins which contaminate the final preparation if this step is omitted. If MiMi-CK is prepared using only Agarose-Hexane-ADP, the resulting enzyme has a reddish color and low specific activity (data not shown). Deoxycholate is included in the initial steps of this procedure to allow the binding of the enzyme to the dye ligand column in Triton X-100 (37). Several initial attempts to purify chicken ventricle MiMi-CK resulted in a preparation which was only approximately 85% pure and had a low specific activity. In an attempt to increase the specific activity, several different chromatographic matricies, which were used in other 116 MiMi-CK preparations, were assayed. These included: carboxymethyl Sephadex (13), phenyl Sepharose CL-AB (AA), Cibacron Blue-agarose (35), amino-heptane Sepharose, Sephadex G-75 and Biogel P-200. These procedures either failed to increase the specific activity of the final enzyme or resulted in the loss of more than 80% of the enzyme. The success of the Agarose-Hexane-ADP column depends on the use of a transition state analog complex obtained by mixing ADP, KNO3 and creatine in the presence of MgC12. Here the nitrate ion mimics the planar transition state phosphate configuration and a strong complex is formed (36). The ADP column permits immobilization of the enzyme to the column under conditions which prevent the binding of two major contaminants (Figure IV-2). Note that a large amount (25%) of the activity is lost during this step. Although the reason for this loss is unexplained, it is probable that the released enzyme has an altered active site because it fails to bind under conditions which promote the formation of a transition state analog. A comparison of the characteristics of MiMi-CK from four different animal mitochondria is presented in table IV’S. The molecular weight and number of subunits are the same for all four species but the specific activity varies considerably between species. This variation may be, in part, due to the different methods used to measure activity and protein concentration. However the Specific activities measured using the commerical kits are more constant. The pI values for the chicken heart is similar to that of the dog heart (as estimated by the direction of enzyme migration at pH 8.8 on cellulose acetate electrophoresis) but much higher than the human heart enzyme. The kinetic constants for the chicken heart enzyme are closer to those 117 .cocfiecocmc p0: .a.z .mpmdpmnsm Locuo 0:» co mcofipmcpcmocoo mmemcsomm pm cocznmoe..x .w.o ma pm szv concoe fixammom an cocfiecouou .n .2.» ma am > socc oocHSLOSoU .H .m.o ma AeocoofinHmov xmameIxao mean: cocfisLOSoc .2 .w.o ma .Axocozv mmmmw1>sxxo mcfims coccHELococ .m .w.o :a Amswfimv xmmmm HmH> OHmCAm xmo mean: coccflscouoc .0 .mamm ouHemHmcomAHoa mom :0 000.2: «0 mufim pacsnzm m mesmmms can oomlm Hzcomcaom :0 cofiumcuafim an 000.00 1 L: 0.0L000L Azkw hHm mm oomco .0 .Ambu mLooom cam 83 00196105 mpcgom 80.5 .0 2m: .3 00 3mm 59C .0 ASFV .Hm pm scam 0cm Azev .Hm so oomco scan .0 .auspm mac» soda memo .m .0002 .a.z mm Fm.o ms m we m Haas .a.z 1 zs m.s 1. ze.w 1_ 25.0, to .a.z 2m m_o.o 2m m_.o ms Pmoso isoemz .Q.z 2s om.o :2 s._ :8 mm_.o s muaeamz xmozam> x acocmaa< .s.z .a.z e._2 s + _m owma nAo omv oas HAooomv cam Aooamv 02m lo ems __2 . wees me lo ems em. maaaasaaa to c 0 m u o H .o.z .a.z Aces m_ mom mamocscsa sic . . soa>aco< oHCHooam s A .o.z w.o a A Ha m m m m moaascsa ooo.sm ooo.se oooo.sw ooo.em 2: cosmor mom oocmo: Loom cacao: amen: mpcmom coxofino .moocsom monLm> Eocm uofluwcza xQIHsz ho moHpmficmpommeo "m1>H manme 118 measured for the beef heart enzyme and other non purified MiMi-CK enzymes (A) than the human heart enzyme which appears to have values similar to the MM isozyme (11). References 1. 7. 9. 10. Neureimer, D. (1981) $2 Creatine Kinase Isoenzymes (Lang, H., ed.). PP. 85-109, Springer-Verlag, New York. Dawson, 0., Eppenberger, H.M., and Kaplan, N.O. (1967) g. 2191. 99293 2A2, 210-217. Jacobs, H., Heldt, H.W., and Klingenberg, M. (196A) Biochem. Biophys. Egg. EQMM. 16, 516-521. Jacobus, W.E., and Lehninger, A.L. (1973) 2. Biol. Chem. 2A8, A803-A810. Ingwall, J.S., Kramer, M.F., and Friedman, W.F. (1980) 12 Heart Creatine Kinase: The Integration of Isozymes for Energy Distribution (Jacobus, W.E., and Ingwall, J.S., eds.). pp. 9-17. Williams and Wilkins, Baltimore, Md.. Bennett, V. D., Hall, N., DeLuca, M., and Suelter, C.H. (1985) 2222' Biochem. Biophys. 2A0, 380-396. Scholte, H.R., Weigers, P.J., and Wit-Peeters, B.M., (1973) Biochim. Biophys. M223 2A1, 76A-773. Baba, N., Kim, S., and Farrell, E.C., (1976) 3. M91. 9211. Cardiol. 8, 599-617. Ogunro, E.A., Peters, T.J., Wells, 0., and Hearse, D.J.. (1979) Cardiovascular Res. 13, 562-567. Vial, C., Font, B., Goldschmidt, D., and Gautheron, D.C. (1979) Biochem. Biophys. Egg. Comm. 88, 1352-1369. 11. 12. 13. 1H. 15. 16. 17. 18. 19. 20. 21. 22. 23. 2“. 25. 26. 119 Watts, D. C. (1973) lo The Enzymes (Boyer, P.D., ed.), Vol. VIII, pp 38u-H55, Academic Press, N.Y.. Hall, N., Addis, P., and DeLuca, M. (1979) Biochemistry 18, 17MB- 1751. Roberts, R., and Grace, A.M., (1980) 3. Biol. Emom. 255, 2870- 2877. Grace, A.M., Perryman, M.B., and Roberts, R. (1983) g. Biol. EEm. 258, 153116-1 53511. Chegwidden, W.R., Hewett-Emmett, D., and Penny, 8.8. (1985) loo. 3. Biochem. 17. 7M9-752. Korenfeld, C.D., Roman, D.C., and Strauss, A.W., foo. Emoo. ”5, Abstract # 55. Blum, H.E., Deus, B., and Gerok, W. (1983) 3. Biochem. 9", 12M?- 1257. Roberts, R., (1980) Experientia 36, 632-63“. Saks, V.A., Kuznetov, A.V., Kupriyznov, V.V., Miceli, M.V., and Jacobs, W.E. (1985) 3. Biol. Chem. 260, 7757-776“. Hall, N., Addis, P., and DeLuca, M. (1977) Biochem. Biophys. Res. gomm. 76, 950-956. Oliver, I.T. (1955) Biochem. o. 61, 116-122. Rosalki, S.H. (1967) g. 222°.9llfl°.!£9° 69, 696-705. Toth, P.F., Ferguson-Miller, S., and Suelter, C.H. (1986) flooo. Enzymol. 125, 16-27. Udenfriend, S., Stein, 8., Bohlen, P., Dairman, W., Leimgruber, W., and Weigele, M. (1982) Science 178, 871-872. Hall, N., and Deluca, M. (1976) Anal. Biochem. 76, 561-567. Laemmli, U.K. (1970) Nature (London) 227, 680-685. 27. 28. 29. 30. 31. 32. 33. 3M. 35. 36. 37. 38. 39. A0. “1. 142. 120 Gracy, R.W. (1977) Meth. Enzymol. XLVII, 195-20“. Bothwell, M.A., Howlett, G. J., and Schachman, H.K. (1978) g. Biol. Chem. 253. 2073-2077. Pollet, R.J., Haase, E.A., and Standaert, M.L. (1979) g. §l2£° 911313. 2511, 30-33. Nickerson, J.A., and Wells, W.W. (198“) o. E12£° Emom. 259. 11297-1130”. Polakis, P. (1985) Ph.D. Thesis, Michigan State University. Storer, A.C., and Cornish-Bowden, A. (1976) Biochem. o. 159, 1-5. Konigsberg, W.H., and Henderson, L. (1983) Meth. Enzymol. 91, Suelter, C.H., and DeLuca, M. (198A) Anal. Biochem. 135, 112-119. Walliman, T., Zurbriggen, B., and Eppenberger, H.M. (1985) 13 Enzyme (Bachman, C., Colombo, J.P., Eppenberger, H., Greengard, 0., Sperling, 0., and Wiesmann, U., eds.). pp. 226-231, Karger AG, Basel, Switzerland. Milner-White, E.J., and Watts, D.C. (1971) Biochem. o. 122, 727- 7H0. Robinson, J.B., Strottmann, J.M., and Stellwagen, E. (1980), Pmoo. flog. .fl£§Q° Sol. £Q§fll 77. 58A7-5851. Scopes, R.K. (197“) Anal. Biochem. 59, 277-282. Wolf, P. (1983) Anal. Biochem. 129, 1M5-155. Groves, W.E., Davis Jr., E.C., and Sells, B.H. (1968) Anal. Biochem. 22, 195-210. Van Iersel, J., Frank, J., and Duine, J.A. (1985) Anal. Biochem. 151, 196-20”. Lebhertz, H.B., Burke, T., Shackelford, J.E., Strickler, J.E., “3. an. 45. “6. M7. N8. 121 and Wilson, J. (1986) Biochem. B. 233, 51-56. Morris, G.B., Frost, L.C., and Head, L.P. (1985) Biochem. B. 228, 375-381. Weselake, R.J., and Jacobus, H.K. (1983) Clin. Chim. Acta 13”, 357-361. Kawashima, 8., Nomoto, M., Hayashi, M., Inomata, M., Nakamura, M., and Imahori, K. (198”) B. Biochem. 95, 95-101. Dean, B. (1983) Arch. Biochem. Biophys. 227, 15M-163. Eppenberger, H.M., Dawson, D.M., and Kaplan, N.O. (1967) B. Biol. Chem. 2&2, 2OA-209. Rosalki, S. B. (1967) B. Lab. Clin. Med. 69, 696-705. Chapter V Association of Avian Mitochondrial Creatine Kinase with the Inner Mitochondrial Membrane Abstract The stoichiometry and dissociation constant for the binding of homogeneous chicken heart mitochondrial creatine kinase (MiMi-CK) to mitoplasts was examined under a variety of conditions. The effect of salts and substrates on the release of MiMi-CK from mitOplasts suggests that this interaction is ionic in nature. Adriamycin competitively inhibits the binding of MiMi-CK to mitoplasts suggesting that MiMi-CK and adriamycin compete for the same binding site. However fluorescence measurements show that adriamycin binds to MiMi-CK suggesting that this latter process, and not the binding of adriamycin to the IMM, may be responsible for the observed competition between MiMi-CK and adriamycin. Titrating mitoplasts with homogeneous MiMi-CK at different pH values shows that the binding of MiMi-CK to mitoplasts can be described by a pH dependent equilibria involving a group(s) on either the membrane or the enzyme with a pKa of about 6. Extrapolating these titrations to infinite MiMi-CK concentration gives 1u.6 IU bound per nmole cytochrome 333 corresponding to 1.12 moles MiMi-CK/mole cytochrome 223 assuming that pure enzyme has a specific activity of 12A IU/mg. Chicken heart mitochondria contain, after isolation, 2.86 I 0-”2 IU/nmole Cytochrome 233 showing that only 22% of the MiMi-CK 122 123 binding sites are occupied in intact mitochondria. Titrating respiring mitoplasts with carboxyatractyloside gives a value of 3.3 moles ADP/ATP nucleotide translocase per mole cytochrome oo Therefore, chicken 3. heart mitoplasts can bind about 1 mole of MiMi-CK per 3 moles nucleotide translocase maximally; in normal chicken heart mitochondria about 1 mole of MiMi-CK is bound per 13 moles nucleotide translocase. Introduction The mitochondrial isozyme of creatine kinase (MiMi-CK, adenosine-S'-triphosphate creatine phosphotransferase; EC 2.7.3.2)3 is located on the outer surface of the inner mitochondrial membrane (IMM) (1,2) and is demonstrably different from either of the cytosolic forms as shown by lack of antibody cross reactivity and hybridization studies (3). Evidence for the participation of this isozyme in Bessman's creatine phosphate shuttle (”,5) comes from studies showing that both MiMi-CK and the adenine nucleotide translocase have a lower apparent Km value for the nucleotide substrate when both enzymes function in a coupled reaction (6-12). These studies also demonstrate increased rates of creatine phosphate synthesis are at identical solution ATP concentrations when the MiMi-CK reaction is coupled to oxidative phosphorylation as compared to the soluble enyzme. The data suggest that MiMi-CK, presumably bound near the adenine nucleotide translocase on the IMM, may have preferential access to the ATP produced by 3f Abbreviations used: AP5A; P'.P5-Di(adenosine-5')pentaphosphate, BSA; bovine serum albumin, DTT; dithiothreitol, EGTA; ethyleneglycol-bis-(Z- aminoethyl ether) N,N,N',N'-tetraacetic acid, IMM; inner mitochondrial membrane, MiMi-CK; mitochondrial creatine kinase, MES; 2-(N- morpholino)ethanesulfonic acid, MOPS; 3-(N-morpholino)propanesulfonic acid. 12A oxidative phosphorylation. The ADP produced by the creatine kinase reaction is then taken up into the mitochondrial matrix via the adenine nucleotide translocase and the creatine phosphate diffuses out of the mitochondria. This sequential utilization of substrates may have an important role in muscle bioenergetics. Dystrophic chicken breast muscle mitochondria contain significantly lower amounts of MiMi-CK when compared to age-matched control birds (1“). Interestingly, these mitochondria show both a decreased rate of creatine phosphate production and a lower ATP trapping efficiency. To ascertain the quantitative significance of the reduced level of MiMi-CK on the functional coupling between MiMi-CK and the nucleotide translocase in muscle mitochondria, we examined the function of MiMi-CK in chicken heart mitochondria because they contain only 5% as much MiMi-CK as normal breast muscle mitochondria (11). Prior to undertaking this study, however, it is necessary to determine the conditions which influence the binding of MiMi-CK to the IMM. Previous studies of this interaction are either difficult to interpret (13, 15-18) or not sufficiently complete to allow a detailed description of the optimal binding conditions (1“, 19). Although results of the studies of the release of MiMi-CK from whole mitochondria (15-17) are difficult to interpret in light of the fact that MiMi-CK interacts with the outer surface of the IMM, MiMi-CK can be solubilized from either whole mitochondria or from mitoplast preparations by increasing ionic strength (12, 1A-17). Other evidence provided by Muller oi El (18) also shows that adriamycin, a cationic lipid (20-22), releases MiMi-CK presumably from the negatively charged cardiolipin head groups on the inner membrane (23). Schlame and 125 Augustin (19) confirmed the binding site by looking at the release of MiMi-CK induced by the action of phospholipases. This paper reports the effects of salts, enzyme substrates, pH, and adriamycin on the binding of MiMi-CK to IMM. Studying the binding of homogeneous MiMi-CK provides a direct determination of the effect of pH and adriamycin on the stoichiometry and dissociation constant for its interaction with IMM. The effect of adriamycin on the binding of MiMi-CK is more complex than previously reported (18,19). Materials and Methods Preparation of Avian Heart Mitoplasts: Mitochondria were prepared daily from avian hearts as previously decribed (2A). The final mitochondrial pellet in 1 mL total volume of S/M buffer (75 mm_sucrose, 225 m! mannitol, 0.1 mm EGTA, 0.2% fatty acid free BSA) gives a suspension 5-8 um in cytochrome oo3 as determined from the difference between the absorbance of the reduced (602 nm) and the oxidized (630 nm) forms (Ac . 2A mflfjcm-1, 2H). Mitoplasts were prepared fresh each day essentially as described by Allman oi oi (25). Digitonin, 0.6 mg per nmole cytochrome 223' was added dropwise to the resuspended pellet with rapid mixing. The resulting suspension was incubated for 10 minutes on ice, diluted 10 fold with S/M buffer, and centrifuged at 8,000 X g for 10 minutes. The mitoplast pellet was subsequently washed two times with S/M buffer and resuspended in a final volume of 0.85 mL of S/M buffer. Using 0.6 mg digitonin/nmole cytochrome oo results in a 3 mitOplast preparation with less than 5% of the original monoamine oxidase activity and adenylate kinase activity. The malate dehydrogenase activity associated with the mitoplast pellet was 3.95% 126 of the original mitochondrial value and the respiratory control ratios were Z_7 demonstrating intact inner membranes. Mitoplasts contain 2.66 I 0.3“ IU MiMi-CK per nmole cytochrome oo compared to 2.86 :_0.A2 IU 3 in intact chicken heart mitochondria. Purification of MiMi-CK: MiMi-CK was prepared from eight 6-8 week old white leghorn chicken hearts according to the procedure of Chapter IV. The final enzyme solution was dialyzed against 10 mm MOPS, 25 mm 2- mercaptoethanol, 2% glycerol and stored at NOC for greater than A months without noticeable loss of binding ability or catalytic activity provided the 2-mercaptoethanol concentration remains high. Titrating Mitoplasts with MiMi-CK: Unless otherwise indicated, the binding of MiMi-CK to mitoplasts was performed in 75 mm_sucrose, 225 mm mannitol, 10 mm MOPS (pH 7.A), 25 mm 2-mercaptoethanol, 0.2% BSA and 50 um ApSA. Mitoplasts, buffer, and various effectors (final volume 0.1 mL), were mixed in 0.A mL Eppendorf centrifuge tubes and incubated at 30°C for 10 minutes. The tubes were spun at 8,000 X g for 10 minutes at 25°C and the supernatants were immediately removed and assayed for creatine kinase activity. Pelleted activities were not commonly measured but initial experiments demonstrated that the activities of the pellets and supernatants always added up to approximately 100% of the initial activity. TitratingiAdenine Nucleotide Translocase with Carboxyatractyloside: This procedure was performed essentially according to Forman and Wilson (26). From 0 to 8 moles of carboxyatractyloside per mole cytochrome 127 3&3 were added to mitoplasts (approximately 0.1 nmole cytochrome oo3) and incubated 2-3 minutes at 30°C in a 1.75 mL respiration chamber containing 5.8 mm pyruvate, 2.9 mm L-malate, in 225 mm mannitol, 75 mm sucrose, 20 mM inorganic phosphate pH 7.“. Oxygen consumption was initiated by the addition of N00 nmoles ADP. The rates of oxygen consumption (Clarke electrode, Yellow Springs Instruments) were recorded as the rate after the addition of ADP (state 3) minus the rate prior to ADP addition (pre-state 3). Titrating Mitoplasts with Adriamycin: Solutions containing from 8 to 250 um adriamycin were incubated with mitoplasts (0.35 um cytochrome 33 ) in 75 mm sucrose, 225 mm mannitol, 10 mm MOPS (pH 7.“), 25 mm 2- 3 mercaptoethanol, and 0.2% BSA. Stock solutions of adriamycin were prepared fresh each day and proctected from light. Adriamycin concentrations were low enough so that insignificant amounts of the aggregated forms were present (22). The decrease in the concentration of adriamycin in the supernatant and its increase in the pellets was monitored using an e - 13,200 [ii-10m.-1 at “80 nm (28). The pellets were solubilized in Triton X-100 and sonicated for 10 minutes prior to measurement. Both procedures gave a concentration of bound adriamycin which did not deviate by more than 5%. Other Procedures: Protein concentrations were determined by the fluorescamine procedure using BSA as a standard (28). Creatine kinase activities were measured by adding an aliquot of the enzyme solution to 0.3 mL of Sigma CK assay mix and recording the absorbance at 3A0 nm at 300C. Malate dehydrogenase activity was measured in a 1 mL reaction 128 mix containing 0.25 mm NAD+, 0.1 mm malate in 0.1 m Tris buffer pH 8.8 and recording the absorbance changes at 3M0 nm. Previously published procedures were used to monitor monoamine oxidase (29) and N-acetly glucoseaminidase (30) activity. Results Effects of Neutral Salts on the Interaction of MiMi-CK with Mitochondria and Mitoplast Preparations: The data in Figure V-1 shows that 50 mm NaCl releases greater than 95% of the endogeneous MiMi-CK from mitoplasts. However, when intact mitochondria are titrated with NaCl, less than 50% of the enzyme is released at 50 mm NaCl; 80 mm NaCl only releases 60%. The release of monoamine oxidase activity approximately parallels the release of MiMi-CK from intact mitochondria suggesting that the outer membrane interferes with the complete release of MiMi-CK. Cellulose acetate electrophoresis confirms that MiMi-CK is released (data not shown). 1 Figure V-2 and Table V-1 show the effect of other salts on releasing MiMi-CK from mitoplasts. Contrary to previously published results (16) showing that more than 20 min is required to establish an equilibrium between free and mitochondrial bound MiMi-CK at a given salt concentration, the equilibrium position is established in less than 2 minutes with mitoplast preparations. The solubilization process is readily reversible; dialyzing suspensions of mitoplasts with solubilized MiMi-CK in 50 mm phosphate against a buffer without phosphate results in rebinding of the enzyme to the IMM (data not shown). The concentration of several salts causing release of 50% of the total MiMi-CK activity are tabulated in Table V-1: the C50 values Figure V-1: 129 Release of MiMi-CK from mitochondria and mitoplasts. Percent of total enzyme activity found in the supernatant liquid after incubating either mitochondria (0.1M nmole cytochrome aa ) or mitOplasts (0.093 nmole cytochrome aa ) are plotted—a; a function of the NaCl concentation. MTM - CK (o, o) and monoamine oxidase activity (a) were assayed as described in the Methods section. The assay conditions were pH = 7.“, T = 30°C. 130 '00 _ O mitochondria o mitoplasts °/. Free 0 MAO O O 50 - _ /:" 0 :-= -:-!'/ . , O 25 50 75 [NaCl] (mM) 131 are expressed in molarities, the CIS values are expressed as ionic strength. Although the C50 values vary from 50 mm for LiCl to 3.9 mm for sodium perphosphate, the CIS values fall within a much narrower range (20-5“ mm) suggesting that ionic strength is primarily responsible for the release of the enzyme and that the interaction of MiMi-CK with IMM is primarily electrostatic. However, since monovalent salts with lower viscosity B coefficients (KSCN, KCl) are slightly more effective at removing MiMi-CK from the IMM than salts with higher viscosity B coefficients (LiCl, (CH )“NCl) (31) there may be a small 3 non-polar effect in the binding. The data in Table V-2 show that none of the various salts specifically release MiMi-CK from the membrane. Most of the specific activity values of MiMi-CK solubilized by several salts at concentrations equivalent to 2 times the C value are near 3 IU/mg 50 consistent with the lack of a specific ion effect. Effect of Substrates on the Interaction of MiMi-CK with Mitoplasts: The effect of increasing substrate concentrations on the distribution between free and IMM bound MiMi-CK is presented in Figure V-3 and Table V-3. The C50 value for creatine phosphate is similar to the C50 value for sodium phosphate suggesting that the effect of creatine phosphate on releasing MiMi-CK is primarily due to an increase in ionic strength and not to a specific interaction with the enzyme. The lack of a significant effect of creatine, a neutral zwitterionic species which does not contribute to the ionic strength of the medium, on releasing MiMi-CK from mitoplasts is consistent with previous conclusions that electrostatic interactions are primarily responsible for the binding of Figure V-2: Figure V-3: 132 Effect of increasing salt concentrations on the percentage of MiMi-CK released from mitoplasts. Various concentrations of salts were added to mitoplasts (1.0 to 1.6 nm cytochrome FE ) at pH 7.11. Supernatant liquids were assayed for activity after centrifugation as described in the Methods section. Release of MiMi-CK from mitoplasts with increasing substrate concentrations. Conditions are identical to those described in Figure V-2. 133 ole Free 1 1 I I I 100 1- O . O 50 L 0 NaCl . A COCIZ A NOPi . O ch12 ' . ’ I NoPPi O .o 6 . A . . 1 1 ' 0 IO 20 30 40 [8011] (MM) 100 .. o/o F786 50 0 IO 20 3O [substrate] (mM) 13” Table V-1: C and C values for the release of MiMi-CK from m?80plasts by various salts at pH 7.". 33.1.31- ‘3... E_1__3__ (mM) (mM) LiCl 50 :.3 50 (CH )uNCl 37 1 1 37 NaC} 35 :_1 35 KCl 31 :.2 31 RbCl 30 1 1 30 CsCl 30.: 5 30 KSCN 25 :.2 25 M3012 7 1 2 20 CaC12 16 _+_ 2 118 BaC12 '8 .t 2 2!: Na SO” 18 :_1 5A sogium phosphate 1“ :_1 35 sodium pyrophosphate A 1.1 25 Note: C 0 values are the salt concentrations at which 50% of the endogengous MiMi-CK is present in the supernatant liquid after centrifugation (see Methods section). The C values are the ionic strength of the salt solution at the 050 val£§. 135 MiMi-CK with IMM. Adenine nucleotides also appear to promote release of MiMi-CK by increasing ionic strength. The nucleotides alone solubilize MiMi-CK at lower concentrations than the 1:1 nucleotide:MgCl2 mixture even though the free nucleotide binds the enzyme with a much lower affinity (32). Furthermore, solutions containing 1:1 mixtures of nucleotide and MgCl2 have lower ionic strengths than free nucleotide (33). Stoichiometry and Dissociation Constant for the Interaction of MiMi-CK with Mitoplasts: Before determining the binding parameters for the interaction of MiMi-CK with mitoplasts, it was necessary to show that MiMi-CK interacts with mitoplasts and not with possible contaminating organelles such as lysosomes. To do this, mixtures of mitochondria or mitoplasts and exogenous MiMi-CK were centrifuged on a continuous sucrose gradient in order to separate the various components of the organelle preparation. As shown in Figure V-Na, MiMi-CK activity follows the peak of malate dehydrogenase activity (mitochondrial marker) and not the lysosomal marker N-acetyl glucosaminidase; the ratio of CK activity to malate dehydrogenase activity (MDH) is constant across the peak fractions. Figure V-ub shows the sedimentation profile for mitoplasts. Note that protein, malate dehydrogenase and creatine kinase migrate in the sucrose gradient as a single band. Also note that N-acetyl glucosaminidase activity is very low in the mitoplast preparations showing that digitonin treatment lyses the lysosomes as was previously reported (3“). A study of the interaction of purified MiMi-CK with mitoplasts as a function of pH gave the results shown in Figure V-5. Here the 136 Table V-2: Specific activity of MiMi-CK released by various salts at 2 X C concentrations. 50 Salt Concentrationa Specific activity (mfl) (IU/mg) NaCl 60 2.57 KCl 60 3.12 MgCl2 1H 3.10 BaCl 32 1.1“ sodium phosphate 27 3.21 sodium pyrophosphate 8 3.21 Na SO 36 2.61 creatgne phosphate N0 2.79 MgADPb 6 1.97 MgATP C 7 2.08 sodium deoxycholate 0.5% 0.83 Note: a. The salt concentrations in these experiments were equal to 2 times the C5 value. b. MgCl and nucleotide were mixed in a 1:1 molar ratio. 0. godiun deoxycholage at 0.5% solubilizes the mitoplasts so that the specific activity is based on the total mitoplast protein. 137 Table V-3: and C values for the release of MiMi-CK from C m?goplasts by enzyme substrates. 333333333 3’3in 329.. 32$. (mg) (mg) ADP -3 3.1 :_1 1 . ATP —u 1.3 1 1 2 .11 creatine 0 ???? --- creatine phosphate -1.8 21 :_3 50.7 MgADPa several species 2.1 1.1 1H.” MgATP several species 3.5 :_1 1H.5 Note: 050 values are the concentrations at which 50% of the endogeneous MiMi-CK 13 present in the supernatant liquid after centrifugation (see Methods section). C values are the ionic strength of the salt solution at the C 0 value. a. MgCl and nucleotides were mixed in a 1:1 ratio. The cgncentration of MgADP and MgATP at the C 0 values are calculated from the equilibrium constants for these specigs (33). The C values represent the total ionic strength of the solution which contains several salt species. Figure V-A: Figure V-5: 138 Association of MiMi-CK with mitochondrial membranes. MiMi-CK (97 nmoles) was added to (A) mitochondria or (B) mitoplasts (2. 6 nmole cytochrome aa ) in 75 mm sucrose, 225 mM mannitol, 0.2% BSA, 10 mM M038, and 2mM DTT (pH 7.") In a total volume of 1.0 mL. This solution was layered over a 26 mL 30-65%'sucr03e (w/v) gradient (over a 3 mL 70% (w/v) sucrose cushion). Samples were then centrifuged at 25,000 rpm in a Spinco SW 25.1 rotor for 3 hours at 80 C. Fractions 1. 5 mL were collected and assayed for MiMi-CK (o), malate dehydrogenase (o), lysosomal N- acetyl-B-d-glucosaminidase (A) and protein (A). The ratio of MiMi-CK to malate dehydrogenase activity is shown over the peak fractions. Binding of purified MiMi-CK to mitoplasts: Klotz plot. Varying amounts of MiMi-CK (final concentration 19 to 600 nM) were added to a fixed concentration of mitoplasts (0.29 uM to O. 113 11M cytochrome aa depending on the experiment) at a final pH value of 5.9; 0, pH 6.2; 0, pH 6. H; A,pH 6. 8, A, pH 7.1; I, pH 7. h;'u, or pH 7.9; V. The buffer consisted of 7 mm MOPS, 7 mm MES, 7 mm Tris, 75 mm surcose, 225 mm mannitol, 0. 2% ABSA, 2% glycerol, 25 mm 2- mercaptoethanol, and 50 uM ApA Other conditions are described in the Methods sect§on. The lines are theoretical curves drawn using parameters obtained from regression analysis (38). 139 9333 A3235 p p ‘Ilvw \.I M. In. 3 e o. m N .c... n m m .m ._ m L1 0 ‘IB 5. 0.0. O I. . . :23: 0.8 N F p . _ 1.. m 5 0.6. 5 o o 395:: HIDE“. m 0 5 4 2 0. 0. O O. Q m n 4 SE\D.EV non—542”. m 1&0 number of moles of MiMi-CK bound to the IMM per mole cytochrome oxidase (3), divided by the maximum number of moles of MiMi-CK bound per mole cytochrome oxidase (n) is plotted as a function of the log of the free enzyme concentration according to Equation V-1. [v-1] V/n = [MiMi-CK] /([MiMi-CK] + Kgpp) free free The value of n was determined for each pH value from a weighted linear regression analysis of the data using a computer program (Appendix A). Several points are evident from Figure V-5. First, the binding is described by a rectangular hyperbola. Second, the lower the pH value, the greater the affinity of MiMi-CK for the IMM. Third, for any given pH value, when free MiMi-CK is above 330 nm, the amount of enzyme bound is no longer described by Equation V-1: this altered binding behavior is presunably due to the highly cooperative concentration dependent polymerization of the enzyme (35). The increased affinity of MiMi-CK for the IMM at lower pH values can be mathematically described by the simplified equilibria of Equation V-2 (see 36). In this system, protonation of A results in an altered affinity for B. A \ \ AB K \ 1 \ + + \ \ B AH+ \ \ AH+B K 191 In Equation V-2 K1 and K3 are the dissociation constants for B binding to A in the unprotonated (A) or protonated (AH+) form and K2 and Kn represent the dissociation constants for protons (H+) binding to A in the free (A) or bound (AB) form. Solving Equation V-2 gives a general equation which relates the bound and free concentrations of A by a function similar to that of Equation V-1 where the Kapp value is given D by Equation V-3. + + a [A + AH ][B] - K3 (K2 + [H J) [v-3] KDpp - [AB + AH+BJ (Kn + [H111 Because of the complexity of the MiMi-CKleM system, it is difficult to determine whether MiMi-CK, or the IMM binding site, or both interactants are protonated. Irrespective of the mechanism of the protonation reaction, the binding equilibria can be described by Equation V-2. In order to determine the four equilibria governing the binding reaction, the pH dependent apparent K values (Kgpp), are D plotted in Figure V-6a as a function of the proton concentration. The error bars in Figure V-6 represent :.1 standard deviation obtained from the least squares fit of the data in Figure V-5 (see legend to Figure v-5). Estimating a value for K from the limiting pK values at high 1 D pH values is technically difficult because of the weak binding at high pH values. However, values for K2 and K3 can be estimated using Equation V-A which is derived from Equation V-3 by assuming [H+]>>Ku. - app [v u] KD . K3 , K2 K3/[H+] a plot of Equation V-A, given in Figure V-6b, demonstrates the validity 142 Figure V-6: Effect of pH on the apparent K to mitoplasts. D value for MiMi-CK binding A: Conditions are as described in Figure V-N. Error bars represent + 1 standard deviation of the K value obtained from a computer analysis of the reciproca? plot (36). The curve is theoretically derived as described in the text. Inset: Release of endogeneous MiMi-CK from mitoplasts as a function of pH. Mitoplasts (0.1“ nmole of cytochrome oo3) were incubated in a medium consisting of 75 mm sucrose, 225 mm mannitol, 0.2% BSA, 2% glycerol, 25 mm 2- mercaptoethanol, 50 um Ap A, 7 mm MOPS, 7 mm MES, 7 mm Tris, 35 mm NaCl and maingained at a constant ionic strength by adding NaCl. This mixture was incubated at 30 C for 10 minutes, centrifuged, and the supernatant activity assayed. B: Plot of the log of the Kapp value against the reciprocal of the hydrogen on concentration used to determine the values of K and K . See the text for 2 3 details. 143 pKopp 7 F' 9‘ Free “ D 100 I I 6 r \\ '1 a - \\\ 5 r ‘-: ° ; . . 8 PH 4 l l l l 1 ' 4 6 8 10 pH I l 100 - .. K3”(nM) 50 L - 0 ' 1 o 5 'IO 15 1/[H‘1 1m" x 10'6) 1AA of this assumption up to a pH value of 7.1. Linear regression analysis gives pK2 - 6 :_0.65 and K3 - 8 :.2 nm. Estimates of the pKu and K 1 values are obtained by a nonlinear iterative computer fit of the 1 results using the values for pK and K3. The theoretical line of 2 Figure V-6a is drawn using a value of pKu - 9, pK2 - 6 and K - 8 nm. 3 A value of K1 - 8 um can then be obtained from the linkage function: 2 3? Confirmation of the mooel given by Equation V-2 comes from an K K 111'K experiment in which the distribution between free and bound MiMi-CK is measured as a function of the pH of the medium in the presence of 35 mm NaCl (see inset, Figure V-6a). Note that the percent of the total enzyme in the supernatant (% Free) decreases with decreasing pH values in agreement with Figure V-5. The value of Kgpp for the binding of MiMi-CK in the presence of 35 mm NaCl at pH 7.N can be calculated using Equation V-2 and the concentration of free MiMi-CK when 50% of the total enzyme is bound (see Figure V-2). Assuming that the K and K 1 3 but not DKZ and pKa are affected by 35 mm NaCl gives a value of K3 - A5 nB under these conditions in contrast to 8 nm at 0 mm NaCl. The theoretical curve shown in the inset of Figure V-6a is drawn using K3 - 45 nm, pK2 - 6 and pKu - 9. Regression analysis of the data of Figure v-5 to Equation V-2 shows that 1u.6 I.1 IU of MiMi-CK is bound per nmole cytochrome 233 on the inner membrane. Assuming that pure MiMi-CK has a specific activity of about 12“ IU/mg (Chapter IV) indicates that approximately 0.92 moles M1M1“CK/MO18 cytochrome 223 can be bound maximally. Preparations of intact heart mitochondria contain approximately 2.86 IU MiMi-CK/nmole cytochrome 223 indicating that only 20% of the MiMi-CK binding sites /_‘\ 1 \ K 1A5 are occupied. Carboxyatractyloside titration of respiring mitoplasts (see Materials section) gives a value of 3.3 moles nucleotide translocase per mole cytochrome 333 a value which has been found by others (”3). Therefore at infinite concentrations of MiMi-CK, about 1 mole of MiMi-CK is bound per 3.6 moles nucleotide translocase on the IMM. Intact chicken heart mitochondria contain about 1 mole MiMi-CK per 13 moles nucleotide translocase. If the specific activity in Chapter IV is low, then the ratio of MiMi-CK to nucleotide translocase would be lower. Effect of adriamycin: Adding adriamycin to mitoplasts with endogeneous MiMi-CK releases the enzyme associated with the IMM in agreement with previously published reports (18, 19, 23). The data in Figure V-7 suggest that adriamycin and MiMi-CK compete directly for the same IMM binding site. Further investigation of this interaction, however, shows that the reaction mechanism is not as simple as previously suggested (18, 19, 23). Adriamycin inhibits the binding of MiMi-CK with a K1 of 60 um. This binding constant is obtained by correcting the total adriamycin concentration for the amount bound to mitoplasts. Adriamycin binding to mitoplasts has a K value of 116 um and a maximum D of 737 :_15A moles/mole cytochrome 333 (data not shown). This latter stoichiometry is approximately 8 times the number of cardiolipin sites available on the IMM (Appendix A) suggesting that adriamycin dissolves in the IMM (38) or binds to integral membrane proteins. In addition, adding adriamycin to MiMi-CK results in an increase in the fluorescence of adriamycin consistent with its binding to MiMi-CK with a KD - 12 :_5 um (Figure V-8). Binding adriamycin, presumably to a nonpolar site on 1116 Figure V-7: Effect of adriamycin on the binding of purified MiMi-CK to mitoplasts. Mitoplasts (0.25 um in cytochrome oo ) were incubated in the presence of 0-200 um adriamycin 3nd titrated with MiMi-CK (final concentration varied from 19 to 333 nm). The buffer was 10 mm MES (pH 6.8), 225 mm mannitol, 75‘mm sucrose, 0.2% BSA, 2% glycerol, 25 mm 2-mercaptoethanol and 50 u! ApSA. Inset shows a plot of ratio Kapp/K versus adriamycin concentration. The slope of @he lTne gives K1 - 60 um. The total concentration of adriamycin is as follows: 0, 0 um, o, 20 um, A, 50 um, A, 100 um, I, 200 um, and o, 500 pg. Note that the curve for 500 um adriamycin was fit by eye. Figure V-8: Binding of Adriamycin to MiMi-CK. Ordinate shows reciprocal change in fluorescence (arbitrary units) and the abscissa shows the log of the adriamycin concentration (um). Approximately 13 nm MiMi- CK was incubated with 1 to 50'um adriamycin and the fluorescence recorded at 580 nm after excitation at A80 nm (39). The fluorescence response represents the difference between the control tubes (without MiMi-CK) and the sample tubes (with MiMi-CK). The different symbols represent different experiments. See the text for additional details. ' 147 10 D I 8 D O 100 200 300 l/Bound 6 Free [adriamycin] (11!) (nmole CK Inmole 9331 4 , ° A e 2 ’. /' / 1 1 41025 . 0.025 005 1/ Free (11!!) 1.0 0.5 1.0 1.5 2.0 log [Adriamycin] (11M) 148 Figome V-9: Swelling of mitoplasts at reduced osmolalities. Mitoplasts (0.A08 um in cytochrome oo ) were incubated in various dilutions of 75 mM sucrose, 225 mM mannitol, 10 mM MOPS, 2 mm DTT (pH 7.11) 13 a total volune-of 0.7 mi. and " the change in absorbance was monitored at 5A6 nm according to Packer (40). Malate dehydrogenase activity was assayed as described in the Methods section. 149 I I l | r j " '00 Relative “I. MDH A 1 released 546 (0) (0) 0.5 *- - 50 W J.) O 1 i l o . O 100 200 300 osmotic strength (m osmol) 150 MiMi-CK, increases its fluorescence by analogy with the increase in daunomycin fluorescence in nonpolar methanol solutions (39). This solvent effect is confirmed for adriamycin which has the same fluorophore as daunomycin (data not shown). Effect of osmotic strength: Previous studies of the release of MiMi-CK from mitochondria suggest that decreasing osmotic strength solubilizes the enzyme (16, 17). Similar effects are not observed with chicken heart mitoplasts. Decreasing the osmotic strength of the support medium results in an expansion of the mitoplasts as shown by lower A5“6 values (Figure v-9)(u0). Note that an insignificant amount of malate dehydrogenase was released showing that the expansion was not sufficient to rupture the inner membrane. When mitoplasts were titrated with MiMi-CK at three different osmotic strengths, indicated app and the maximal by the arrows in Figure V-9, identical values for KD number of binding sites per mole cytochrome oo were obtained (data not 3 shown). Therefore, decreasing the osmotic strength does not appear to affect the total number of available binding sites for these mitoplast preparations. Discussion Recent reports from other laboratories (7. 9) on the coupling of mitochondrial oxidative phosphorylation with MiMi-CK activity show that both MiMi-CK and nucleotide translocase preferentially accept substrates from each other even when the outer membrane is removed (12). The present study was initiated in order to obtain a more quantitative assessment of this functional coupling. However, before 151 this could be initiated, it was necessary to determine the effect of buffer salts, substrates and ionic strength on the stoichiometry and dissociation constant for the binding of MiMi-CK to IMM under a variety of conditions. The interaction between MiMi-CK and the IMM is primarily due to electrostatic forces as illustrated by the following results. First, at pH 7.", relatively low salt concentrations (§_50 mm ionic strength) release the enzyme in a reversible manner. The release of enzyme is nearly independent of the ion as shown by the similarities in the ionic strengths required to release 50% of the total MiMi-CK activity (Table V-1). Second, the concentration of ionic substrates or nucleotides causing the release of 50% of the MiMi-CK from the IMM are in the same ionic strength range as the neutral salts. When expressed in molarities, the values range from 1.3 mm for ADP to 50 mm for LiCl. Third, the neutral zwitterionic substrate creatine is ineffective in causing release of the enzyme even though creatine phosphate has a 050 of 21 mm. MiMi-CK released from the IMM by ionic substrates or neutral salts at their 050 values has nearly the same specific activity (Table V-2) indicating the lack of a specific substrate or ion effect. Although the concentrations of salts in the range of the C50 values given in Tables 1 and 3 exert little effect on the bulk phase water structure (A1, “2). the C50 values of the simple 1:1 salts are not identical but vary from 50 to 25 mm and follow the order LiCl>NaCl>KCL>RbCl-CsCl>KSCN. This order is the same as the order of their viscosity B coefficients (31) suggesting that the differences between the C50 values reflect a hydrophobic component to the interaction. The differences between the salts may be a reflection of 152 their ability to penetrate non-polar areas of the complex and disrupt ionic interactions (A1, A2). Studying the binding of homogeneous MiMi-CK to mitoplasts under other conditions revealed a strong pH dependence: MiMi-CK binds approximately 1000 times more strongly at pH 6 than at pH 9. Although the data can be modeled by a simple protonation scheme (Equation V-2), the process is almost certainly more complex. The electrostatic nature of the binding reaction suggests that the MiMi-CK, which is positively charged at pH 7.A as evidenced by its migration to the cathode in an electric field at pH 8.8, is the protonated species. A decreased pH value would either neutralize a negative charge or result in an extra positive charge on the protein. Neutralizing a negative charge or producing a positive charge on the IMM would not be expected to increase its affinity for MiMi-CK. Previous studies showed that adriamycin releases MiMi-CK from IMM (23) and that MiMi-CK binds to vesicles containing cardiolipin (18). Hawever we were not able to demonstrate the binding of homogeneous MiMi-CK to either asolectin or cardiolipin:phosphatidyl choline (1:3, w/w) vesicles under conditions where mitoplasts bind MiMi-CK. This result suggested that adriamycin may not prevent binding of MiMi-CK by interacting with cardiolipin but that other mechanisms may account for the competitive inhibition with adriamycin. Further investigation showed that the fluorescence of adriamycin increases when added to MiMi-CK, similar to its response when placed in a hydrophobic medium (39). Assuming that the fluorescence enhancement is due to an adriamycin:MiMi-CK complex gave a K - 12 um (Figure V-8). This value D is smaller than the the K1 for adriamycin inhibition of MiMi-CK binding 153 to the IMM (60 um) but the latter value may reflect the concentration of MiMi-CK binding sites on the IMM in the assay mixture. The binding of adriamycin to MiMi-CK and the lack of binding of enzyme to cardiolipin vesicles suggests that the inhibition by adriamycin is not due to its binding to cardiolipin in the IMM. However, whether MiMi-CK binds to a protein of other charged groups in the membrane cannot be discerned by these data. Two lines of evidence also suggest that, contrary to previous suggstions (A, 5), MiMi-CK does not interact with ADP/ATP nucleotide translocase. First, only 0.3A moles of MiMi-CK bind maximally per mole of nucleotide translocase and second, adding up to 5 moles of atractyloside or carboxyatractyloside per mole of nucleotide translocase does not release MiMi-CK from IMM (data not shown). The stoichiometry for the binding of MiMi-CK to IMM is not consistent with that recently reported by Kuznetov and Saks (A3). Using an oxidized ADP analog affinity label they report that an equivalent of 1 mole MiMi-CK per mole of nucleotide translocase is present in intact chicken heart mitochondria. Using their value of 0.8 units MiMi-CK per mg mitochondrial protein, which is in agreement with our results (Table V-2), gives a specific activity for homogeneous MiMi-CK of 8 IU/mg. This value is much lower than those reported by others (33) and the value of 12A IU/mg which we find (Chapter IV). Perhaps other proteins in the IMM bind the ADP affinity label. Acknowledgments We gratefully acknowledge the assistance of the Department of Animal Science, Michigan State University, and particularly Dr. R. Balander for supplying the chickens. We also acknowledge Dr. J. E. 15A Wilson for helpful suggestions in writing this paper. References 1. 10. 11. 12. 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Robinson, J.B., Strottmann, J.M., and Stellwagen, E. (1981) m. ya_t. _A_c_a_o. _S_c_i_. (_usi) 78, 2287-2291. Watts, D. C. (1973) im The Enzymes (Boyer, P.D., ed.), Vol. VIII, pp. 38A-A55, Academic Press, N.Y.. Storer, A.C., and Cornish-Bowden, A. (1976) Biochem. B. 159, 1- 5. Kun, E., Kirsten, E., and Piper, W.N. (1982) Meth. Enzymol. LV, 115-118. Hall, N., Addis, P., and DeLuca, M. (1977) Biochem. Bioohys. Boo. Bomm. 76, 950-956. Cleland, W.W. (1978) Adv. Enzymol. A5, 326-328. Krebs, J.J.R., Hauser, H., and Carafoli, E. (1979) B. Biol. Chem. 25A, 5308-5316. Fiallo, M.M.L., and Garnier-Suillerot, A. (1986) Biochim. Bioohys. logo 85A, 1A3-1A6. Goldman, R., Facchinetti, Bach, D., Raz, A., and Shinitzky, M. (1978) Biochim. Bioohys. logo. 512, 25A-269. Packer, L. (1967) Meth. Enzymol. 10, 685-689. Felgner, P.L., and Wilson, J.E. (1977) Arch. Biochem. Bio s. A2. A3. 1111. 157 182, 282-29A. Von Hippel, P.H.. and Schleich, T. (1969) im Structure and Stability of Biological and Macromolecules (Timasheff, S.N., and Fasman, G.D., eds.). pp. A17-A51, Marcel Dekker, New York. Kuznetov, A.V., and Saks, V.A. (1986) Biochem. Bioohyo. Boo. Bomm. 13A, 359-366. Veech, R.L., Lawson, J.W.R., Cornell, N.W., and Krebs, H.A. (1979) B. Biol. Chem. 25A, 6538-65A7. Chapter VI Preferential Coupling of Mitochondrial Creatine Kinase to Chicken Heart Nucleotide Translocase in Mitochondria and Mit0plasts Abstract Preferential coupling of mitochondrial creatine kinase (MiMi- CK) to nucleotide translocase in chicken heart mitochondrial preparations is demonstrated. Using intact mitochondria to measure the MiMi-CK Km value for MgATP_2 (at saturating creatine) gives a value of 36 um when MiMi-CK is coupled to oxidative phosphorylation. This Km value is three fold lower than that of MiMi-CK measured with mitoplasts under conditions where about 70% of the MiMi-CK is bound to the inner mitochondrial membrane (IMM). The Km value for MgATP.2 is also three fold higher when the soluble enzyme is assayed using pyruvate kinase and lactate dehydrogenase as the coupling enzymes. The nucleotide translocase Km value for ADP decreases from 20 um to 10 um in the presence of 50 mm creatine only when the outer mitochondrial membrane is present. The observed Km differences can be used to calculate the concentration of ATP and ADP under steady state conditions showing that the observed differences in the kinetic constants accurately reflect the enzyme activities of MiMi-CK under the different conditions. These data indicate that, in the chicken heart system, and similar to the rabbit heart system, intact mitochondria are necessary to observe preferential coupling. 158 159 Introduction Mitochondrial creatine kinase (MiMi-CK)3 is localized in the inter membrane space of mitochondria presumably adsorbed to the outer surface of the inner membrane (1,2). Because of this location, several approaches have been used to demonstrate that MiMi-CK has preferential access to ATP generated by oxidative phosphorylation. Saks oi ol. (5,6) and Jacobus and Saks (7) report a 7 - 10 fold lower Km value for MgATP"2 (at finite creatine) when MiMi-CK is coupled to oxidative phosphorylation. Moreadith and Jacobus (8) and Barbour SE ol. (9) also see a two fold lower nucleotide translocase Km value for ADP in the 32P partitioning experiments by Erickson- presence of creatine. Viitanen oi 23° (10, 11) show that up to 50% of the creatine phosphate is derived from ATP generated by oxidative phosphorylation when the bulk ATP concentration is below 0.1 mm. These experiments indicate that at low ATP concentrations, MiMi-CK has a preferential access to ATP produced by oxidative phosphorylation over ATP in the bulk solution. Two different hypotheses have been advanced to account for the preferential coupling of the nucleotide translocase and MiMi-CK. Saks 22.2l' (5, 6, 12), Jacobus and Saks (7), and Jacobus (13) argue that MiMi-CK is bound to the nucleotide translocase such that the rate of ATP utilization by MiMi-CK is faster than its diffusion into the surrounding solution; the nucleotide translocase does not transfer ATP directly to MiMi-CK as atractyloside does not inhibit MiMi-CK activity (23). This hypothesis is supported by Saks §£.El° (6) who show 3. Abbreviations used: Ap A; diadenosine 5' pentaphosphate, IU; 1 umole of substrate converted pe? minute, IMM; inner mitochondrial membrane, MiMi-CK; mitochondrial creatine kinase, MOPS; 3-(N- morpholino)propanesulfonic acid, OMM; outer mitochondrial membrane. 160 preferential coupling in rat heart mitochondria even when the outer mitochondrial membrane is removed by digitonin. Contrary to the data from Jacobus and Saks' laboratories, Erickson-Viitanen oi oi. (10, 11) show preferential coupling only with intact mitochondria. They argue that the outer membrane acts as a partial diffusion barrier limiting the efflux of newly synthesized ATP and influx of medium ATP (11). Using chicken heart mitochondria and mitoplasts, we have reexamined the importance of the outer membrane and the binding of MiMi-CK to the inner membrane to coupling of MiMi-Ck and oxidative phosphorylation. The data show that MiMi-CK is preferentially coupled to oxidative phosphorylation in intact mitochondria but not in mitoplasts, even though 70% of the total MiMi-CK is bound. We are also able to show that normal enzyme coupling kinetic theory (Chapter II and III) can be used to predict the steady state concentration of the intermediates ADP and ATP. Materials and Methods General: All chemicals, enzymes and creatine kinase assay kits were obtained from Sigma Chemical Co. (St. Louis MO) unless otherwise specified. Common laboratory chemicals were reagent grade or better. Chickens were obtained from the Department of Animal Science, Michigan State University as newly hatched chicks and maintained on Chick 00125 (Kent Feed Inc., Muscatine IA). MiMi-CK was purified from chicken heart mitochondria as described in Chapter IV. Its concentration is determined from activity measurements using a specific activity of 12A IU/mg. 161 Preparation of Mitochondria and Mitoplasts: Mitochondria and mit0plasts were prepared as described in Chapter IV. Washed mitochondria were prepared by suspending mitochondria one time in 50 mm NaCl to remove MiMi-CK which may be solubilized under the experimental conditions due to broken outer mitochondrial membranes. Thirty percent of the creatine kinase activity was lost during this wash. Washed mitoplasts were prepared by incubating mitoplasts in 50 mm phosphate, 10 mm Tris (pH 8.0) at 0°C for 7 minutes. This procedure removes approximately 90% of the MiMi-CK and allows the addition of a high mitoplast concentration so that approximately 70% of the MiMi-CK present is bound to the IMM. The concentration of cytochrome oo was 3 determined from the differences in the absorbance of the reduced minus oxidized spectrum at 602 minus 630 nm (As - 2A mm-1 cm-1, 18). Enzyme Assays: Monoamine oxidase and malate dehydrogenase activities were measured as described in Chapter IV. The kinetic constants of homogeneous soluble MiMi-CK were analyzed as previously described in Chapter III. Because the conditions used to determine the kinetic constants of MiMi-CK release the enzyme from the IMM (Chapter V), the kinetic constants of MiMi-CK bound to the IMM in the presence of oligomycin A and of MiMi-CK in intact mitochondria were determined by following the production of ADP by end point assays. The assays were performed by incubating washed mitochondria or washed mitoplasts in 1 mL of 75 mm sucrose, 222 mm mannitol, 0.2% bovine serum albumin, 2.5 mm phosphate, 10 mm MOPS, 2 mm magnesium acetate (pH 7.0, Buffer A). The reaction, which contained varying amounts of creatine, was initiated by 162 adding a 1:1 mixture of magnesium acetate and ATP. At 0.5, 1, and 1.5 minutes, 0.3 mL aliquots were removed and added to 30 uL 3.B trichloroacetic acid. The solutions were centrifuged for 3 minutes at 8,000 X g, and 250 uL of the supernatant was quickly removed and neutralized by adding KOH. To this solution, 678 uL of 50 mm MOPS, A mm MgClz, 0.2 mm EDTA, 100 mm KCl (pH7.0), 10 uL of 0.1 m phospho(enol)pyruvate, and 10 uL of 67 mm NADH were added. The difference in the absorbance before and after the addition of pyruvate kinase and lactate dehydrogenase gives the ADP concentration. The rates of ADP production are linear over the time course of the experiment. Kinetic parameters for MiMi-CK coupled to oxidative phosphorylation were measured using an oxygen electrode apparatus (Yellow Springs Instruments) in Buffer A plus 5 mm pyruvate and 2.5 mm malate. The reaction was initiated by the addition of a 1:1 molar mixture of magnesium acetate and ATP. The state 3 rates are corrected from velocities measured in the absence of creatine. The lag time for the coupled reaction was always less that A minutes. The kinetic parameters of the nucleotide translocase, in the presence and absence of creatine, were determined using an oxygen electrode apparatus with the normal sensitivity increased six fold. The reactions were initiated by adding various amounts of ADP to Buffer A containing 5 mm pyruvate, 2.5 mm malate, and mitochondria or mitoplasts spiked with 0.38 IU MiMi-CK per assay so that the final ratio of enzyme was 3.3 IU/nmole cytochrome oo3, 163 Measuring Creatine Phosphate and ATP Concentrations: An aliquot (1O uL) of 27 mm ATP plus 27 mg magnesium acetate was added to a suspension of mitochondria or mitoplasts in 1.5 mL Buffer A plus 5 mm pyruvate and 2.5 mm malate. At time 0.5, 1.0, 1.5, and 2.0 minutes, a 0.3 mL aliquot was removed and added to 0.3 mL boiling water (19). The samples were allowed to boil for 1 to 2 minutes, cooled on ice for 10 minutes and centrifuged for 3 minutes at 8,000 X g. The supernatants (0.25 mL) were removed and made 1A mm in glucose, 3 mm in magnesium acetate, 25 mm in 2-mercaptoethanol, 20 mm in Hepes, A% (v/v) in glycerol and 0.1 mm in EDTA. The concentration of ATP was calculated from changes in absorbance at 3A0 nm after adding 1 mm NADP+, hexokinase and glucose 6-phosphate dehydrogenase. Creatine phosphate was measured after adding creatine kinase. Theory The coupling of MiMi-CK to the oxidative phosphorylation is a special case of the two coupled enzyme system presented in Scheme 1 of Chapter II. The equations presented in Chapters II and III show that, when the rate of the primary enzyme is constant, the steady state concentration of each intermediate is dependent on the rate and Km value of the coupling enzyme; the higher the coupling enzyme concentration or the lower the coupling enzyme Km value, the lower the intermediate concentration. When MiMi-CK is coupled to oxidative phosphorylation, the substrate and product for translocase (e2 in Figure VI-1) are the product and substrate for MiMi-CK (e1 in Figure VI-1). Consequently the rate of the primary enzyme (v1) cannot be fixed as the reaction tends toward steady state as was done so 16A conveniently for the general coupled enzyme systems of Chapters II and III. The concentration of ADP at any time t can be expressed as [VI—1] d[ADPJ/dt = v - v 1 2 which says that the change in ADP over time is a function of the difference between the rate of ADP formation (v1) and the rate of ADP utilization (v2). Substituting the Michaelis-Menten relationships for these rates, we obtain: [VI-2] d[ADPJ/dt = V1[ATP]/(K1 + [ATP]) - V2[ADP]/(K2 + [ADP]) where V1 and V2 are the maximal velocities of MiMi-CK and oxidative phosphorylation respectively. K1 is the MiMi-CK Km for MgATP-2 in the presence of a finite amount of creatine and K is the measured Km of 2 oxidative phosphorylation for ADP. Since the nucleotide translocase uses only free ADP and MiMi-CK uses only MgATP-2, the constants c and B, which reflect the binding constants of ADP and ATP for magnesium, are defined as: a = 1 - [MgADP-J/[ADPT] B a [MgATP—23/[ATPT] The value 0 thus represents the fraction of ADP which reacts with the translocase and the value 3 represents the fraction of ATP which reacts With MIMI-CK. The values of [ADPT] and [ATPT] represent the total concentrations of ADP and ATP. If we define the total nucleotide 165 Figure VI-1: Coupling the synthesis of creatine phosphate to oxidative phosphorylation through MiMi-CK. e represents MiMi-CK and e 2 represents oxidative phosphorylation. 166 _& .oNo aaos.ux . ~> 0 and nth a5 6 :33. I. _> L0 167 concentration as AT . [ADPT] + [ATPT], Equation VI-2 can be written as Equation VI-3. d[ADP] V1 * 8(AT - [ADP]) V2 * GEADP] dt K1 + BAT - B[ADP] K2 + d[ADP] [VI-3] Solving Equation VI-3, and a similar equation for the rate of change of ATP with respect to time for the steady state condition (d[ADPJ/dt a 0) gives Equations VI-A and VI-5. 2 [VI A] 0 . ¢EADPJSs + [ADPJSS(V1K2/a + v K1/B ATt) A v K /a 2 T 1 2 TV2K1/B 2 [VI 5] O = ¢[ATP]SS [ATPJSS(V1K2/a + V2K1/8 + ATO) + A where 0 a V - V 1 2. Equations VI-A and VI-5 enable one to calculate the concentration of ADP or ATP for any concentration of e1 and e2 provided K1 and K2 are known. Equation VI-A and VI-5 predict that as e2 increases, [ADP]SS increases and [ATP]SS decreases in a non-linear fashion. If V2 - 0, Equations VI-A and VI-5 predict that [ADPss] = 0 T' Also as V2 + w, Equations VI-A and VI-5 predict that [ADPJSS + A and [ATPJSS + 0. and [ATP]SS . A T Results Binding of MiMi-CK to Mitoplasts: In order to assess the preferential coupling of bound MiMi-CK to oxidative phosphorylation, it is necessary to define the conditions for the binding of MiMi-CK to the IMM. To measure the distribution of free and bound MiMi-CK under the conditions of our experiments, homogeneous MiMi-CK was added to mitoplasts and the Figore VI-2: 168 Binding of MiMi-CK to the inner mitochondrial membrane. MiMi- CK (free concentration from A7 to 20A nB) was incubated with mitoplasts (0. 01 nmole cytochrome aa ) for 10 minutes at 300 C in 100 uL of 5 mB pyruvate, 2. 5 —8B malate, 0. 5 mB magnesium acetate, 0. 5 mB ADP, 50 mB creatine, 1O mB MOPS, 25 mB 2-mercaptoethanol, 75 mB sucrose, 225 mB mannitol, 2% bovine serum albumin and 2.5 mB phosphate (pH 7.0). The concentration of free MiMi-CK was measured in the supernatants following centrifugation at 8,000 x g for 10 minutes. The amount of MiMi-CK bound was obtained by substracting the total activity from the free activity. The line was fit with a linear regression program (Appendix A) using the weighting scheme suggested by Wilkinson (22). The value in parentheses was omitted from the analysis as it lies 2 standard deviations outside the calculated line. 169 7:25 8...: No.0 5.0 _ 1.» Amos 205385 U::Om\ _ 170 free and bound concentrations of enzyme plotted according to Equation VI-6 (Figure VI-2) [VI-6] ;/n - [MiMi-CKJF/(KD + [MiMi-CKJF) where [MiMi-CKJF is the free concentration of MiMi-CK and V/n is the fractional saturation of the available binding sites on the IMM. A KD value of 87 1'12 nB and the maximal number of binding sites, 0.87 1 .06 mole MiMi-CK/mole cytochrome oo agree with the values given in Chapter 3 IV. Using these parameters in conjunction with Equation VI-6 allows one to calculate the amount of MiMi-CK bound and the number of binding sites on the IMM which are occupied under the conditions of the steady state experiments. Measuring the kinetic parameters of MiMi-CK requires the presence of 2 mB magnesium acetate; this concentration has a negligible effect on the binding of MiMi-CK to the IMM. However, adding 1 mB magnesium acetate plus ATP (highest substrate concentrations used in the assay) caused a further 10% release of MiMi- CK from the IMM (see Chapter V). Thus, when the MgATP concentration is close to 1 mB, approximately 60% of the total MiMi-CK is bound to the inner mitochondrial membranes under the experimental conditions of Figure VI-3. Kinetic Constants of MiMi-CK: Figures VI-3a and VI-3b present the intercepts and slopes of the primary plots of 1/velocity and 1/[creatine] for MiMi-CK under a variety of conditions as a function of the log (base 10) of the MgATP”2 concentrations. The kinetic constants obtained from these data are presented in Table VI-1 and defined in 171 Equation VI-7. CK: MgATP2 MgAT:-:l?:a :b'\l Cr Vf creatine phosphate [VI-7] CK: MgATP2 > + _ MgADP MgATP2 CK: Cr Measuring the Ka and Ea values using intact mitochondria with the translocase as the coupling enzyme shows that the Ka and Ea values are approximately three fold lower than those measured with mitoplasts, even though 78% of the MiMi-CK is bound to the IMM when mitoplasts are used (compare open and closed squares, Figures VI-3a and VI-3b). The Ka and Ea values for the soluble enzyme are similar to the bound enzyme suggesting that the outer membrane is responsible for the preferential coupling. The effect of the OMM on the kinetic parameters of MiMi-CK is further indicated by measuring the kinetic parameters in the presence of oligomycin A. In this case, respiration is inhibited and the reaction is measured by following ADP concentration over time. The Ka and Ra values for MgATP.2 are six fold higher using intact mitochondria but are identical when mitoplasts are used even though 69% of the MiMi-CK is associated with the IMM. Note that the K and K b D values for creatine are identical for all five conditions (Table VI-1). Kinetic Constants for Oxidative Phosphorylation: Determining the nucleotide translocase Km value for ADP by measuring the respiration rate in the presence of increasing concentrations of creatine as a 172 pm1H> ocsmfim 00 ocomoa 0:0 as no ncoaoaocoo .0002 I I I I I xms 0m . 00, m. . 00F om . 0e_ 0m + 00, 0 + 00m Ame\0H0 > m _u e F n 0 N u 0_ m n 0 P u 0_ Aoeaoeotov as s u 0_ m n.m_ . u 0? m N am 0 n.0m Aocaoeocov 0m . 1 .. . 1. . . 1. . . .. . . 1 .. e No.0 + m_.0 m_.0 + 00.0 No.0 + 0_.0 P0.0 + 20.0 N0.0 . me.0 Amuae manna 173 Figure V-3: Measuring the kinetic parameters for the synthesis of creatine phosphate by MiMi-CK. The figuro are the secondary plots of the slopes (mB*mg*IU .) and intercepts (mg/IU) of the primary plots of 1/velocity versus 1/[creatine]. The curves are obtained from a linear regression analysis (Appendix A) using weighting suggested by Wilkinson (22). In experiments b to e, mitochondria or mitoplasts were incubated for 2-A minutes at 300C in 75 mB sucrose, 225 mB mannitol, 0.2% bovine serum albumin, 2 mB magnesium acetate, 0.1 mB EDTA, 2.5 mB phosphate, and 10 mB MOPS, pH 7.0 (Buffer B). The creatine concentrations were 50 mB, 35 mB, 20 mB, 10 mB, 7 mB, 5 mB, and 2 mB for experiments a, d and e and 50 mB, 20 mB, 10 mB, 5 mB, and 2 mB for experiments b and c. The reactions were initiated by adding a 1:1 molar mixture of magnesium acetate and ATP, the MgATP concentrations are calculated following Storer and Cornish-Bowden (21). The state three rates were subtracted from rates‘measured in the absence of creatine. The bottom axis is used with the data for experiment b. a. (e) MiMi-CK (0.19 IU) was added to 1.0 mL of reaction mix as described in the Methods section.‘ MgATP varied from 63 uB to 6.A mB. b. (I) Washed mitochondria (0.21 nmole cytochrome oo , 0.A IU MiMi-CK) were added to 1.75 mL 92 Buffer B plus 5 _. ' pyruvate and 2.5 mB malate.‘ MgATP varied from 9 uB to 173 11B. c. (a) Washed mitoplasts (2.7 nmole cytochrome oo , 0.36 IU MiMi-CK) were added to 1.75 mL 95 Buffer B plus 5 mB pyruvate and 2.5 mB malate.’ MgATP varied from 36 uB to 790 AB. ' d. (A) Washed mitochondria (0.1 nmole cytochrome a_a , 0.2 IU MiMi-CK) were addeg to 1.0 mL of Buffer B plus A ngmt oligomycin A. MgATP varied from 0.32 mB to 7.9 mB. e. (A) Washed mitoplasts (0.2A nmole cytochrome oo , 0.07 IU MiMi-CK) were addog to 1.0 mL of Buffer B plus A ug/mL oligomycin A. MgATP varied from 36 uB to 790 uB. 174 10 VS,” V.,, 0.5 O I . l -2 -I 0 IogEMgATP'ZJ (mM) I.0 - smax/Sqap 0.5 " O ‘3: - 1 l -2 -1 0 IogEMgATP'ZJ (m!!!) 175 function of ADP shows that it decreases to 10 uB, about half of the Km value measured in the absence of creatine (Figure VI-Ab) in agreement with the results of Barbour oi_ol. (9) and Jacobus oi oi. (20). Removing of the outer membrane abolishes this effect. It is important to note, however, that only 20% of the available IMM binding sites for MiMi-CK are occupied and that MiMi-CK activities are equal to the oxidative phosphorylation rate when mitoplasts are used to determine the translocase Km value for ADP. Rates of Creatine Phosphate Synthesis: Figure VI-5 is a plot of creatine phosphate and ATP concentrations as a function of time. Although the ATPss concentrations are nearly identical for all three cases, the rates of creatine phosphate synthesis (Vobs) are clearly different (see Table VI-2). When MiMi-CK is coupled to oxidative phosphorylation in intact mitochondria, the steady state rate of creatine phosphate synthesis is nearly twice that seen for the soluble enzyme (squares, Figure VI-5) or enzyme bound to mitoplasts (triangles, Figure VI-5). Since the concentration of MiMi-CK is identical in all three cases, the different rates of creatine phosphate synthesis are the result of different apparent Km values for MgATP“2 (measured at 100 mB creatine) under the conditions of the experiment. These apparent Km values can be calculated from the ATP concentrations and the maximal activity of MiMi-CK by inverting the Michaelis-Menten equation as shown below. - app - [VI 8] Km - [ATPJSSUI1 v )/V obs obs 176 Table VI-2: Determination of the MiMi-CK Km value for ATP and the ATP steady state concentration. calculated Effgififf 3995 Efffiaa apparent Km [ATPJSS a. Mitochondria 80 uB/min 150 uB 16 AB 155 u! Mitoplasts ' b. free A6 uB/min 127 uB 117 uB 1A2 uB c. bound 52 mB/min 1A0 uB 99 uB 1A9 uB Note. v is the rate of creatine phosphate synthesis at steady state. 95% apparent K values are calculated according to Equation VI- 8. The value of V1, wRich represents the maximal rate of creatine phsophate synthesis, is 88.6 uB/min under the conditions of the experiment. The [ATP] values are calculated using K1 - apparent K , and the following values: a. V - 17A uM/min, K - 10 uB and [ATP] the absence of creatine - 169 § 3 uB, b_ V - 97 uB/min,K - 20 uBSand [ATP] in the absence of creatine = 167 + 3 uB, c. 26A uB/min, K2 - 20 uB and [ATP] in the absence of creatine = 155V 5 uB. K and V2 are the K value and maximal rate of ATP synthesis measured in Ehe absence 0 creatine. Note that the decreased value of [ATP] (as compared to the initial value of 170 uB ATP) is due to the aegion of an endogeneous ATPase. This lower ATP concentration is taken as the total nucleotide concentration when the calculated [ATP] values are obtained using equation VI-5. The values are measured under the conditions of Figure VI-A. a - 0.766 and B - 0.85A under these conditions (21). ' ‘ 177 Table VI-3: Steady state ADP concentrations for different mitochondrial preparations. Condition K1 [ADP]SS [ADP]SS obs. calc. (1111) (11!) (11M) Mitochondria 36 10.6 _+_ 1 11.9 Mitoplasts ‘ ' ‘ free 125 18.2 I 0.8 15.A : 1.1 6.8 bound 100 6.5 Note. Washed mitochondria (0.11 nmole cytochrome oo , 0.2 IU MiMi-CK), mitoplasts (0. 07 nmole cytochrome aa 0. 2 IU MiMi- -CK), or washed mitoplasts (0.72 nmole cytochrome aa 0. 2 IU MiMi- CK) were incubated in 1. 5 mL 75 mB sucrose, 225 mB mann§tol, 0. 2% bovine serum albumin, 0. 5 mB magnesium acetate, 10 mB MOPS, 2. 5 mB phosphate, 5 mB 8- hydroxybutyrate, 100 mB creatine (pH 7. 0) for 2 minutes prior to addition of 80 uB ADP. The respiration of washed mitoplasts was inhibited to 30% of normal by adding 3.1 mole carboxyatractlyoside per mole cytochrome aa The values represent the mean of two determinations. T60 [ADP]SS values were calculated vfrom Equation VI-A‘ using the following values: V1 = AA uB/min, and a. - 57 uB/min,K 10 uB, and [ADP] in the absence of creatine - 1 i_v.3 uB, b. 36 uB/min,K - 20 pi, and [ADP]SS in the absence of creatine - 2V + .7 uB, a. 86 uB/min,K - 20 uB, and [ADP]s in the absence of creatine - 3 + .A uB. a - 0. 766 and B - 0. 85A under sEhese conditions (21). The definitions of V V , and K2 are presented in the legend to Table VI-2 and in the Theory section. 178 where V and Kipp are the Vm 1 and apparent Km for MiMi-CK measured in ax the presence of 100 mB creatine. The value of V1, reported in the legend to Table VI-2, is the maximal rate of creatine phosphate synthesis measured under the conditions of the experiment. The apparent Km values are presented in Table V192. Using the apparent Km values determined above, the V1 values for the system, and the rate and Km value for oxidative phosphorylation measured for this experiment (V2 and K2 values, see Theory), permits us to calculate the steady state ATP concentrations using Equation VI-S. The results of this calculation, and the kinetic constants for the two enzymes are presented in Table VI-2 as well. These values are obtained by assuming that the value of ATPSS in the absence of creatine is the total nucleotide concentration. This latter assumption corrects for the presence of an ATPase. Although the calculated ATPSS values are slightly higher than the measured values, they agree well with one another. Measuring ADP Concentrations: In order to measure the effect of different kinetic constants on the steady state concentration of ADP, the three different mitochondrial preparations were allowed to go into steady state and the concentration of ADP measured. The establishment of steady state is confirmed by measurement of an identical ADP concentration at 5 and 6 minutes after the reaction is initiated. The [ADP]SS concentrations presented in Table VI-3 have been subtracted from those measured in the absence of creatine, and thus represent the steady state concentrations due to the activity of MiMi-CK and not to an ATPase activity (see legend to Table VI-3). Equation VI-A is used Figure VI-A: Figure VI-5: 179 Effect of increasing creatine concentrations on the V max and Km for the nucleotide translocase. a. Mitochondria (0.021 nmole cytochrome aa 0.062 IU MiMi- CK) or mitoplasts (0.018 nmole cytoch ome aa 0.058 IU MiMi- CK) plus 0. 38 IU MiMi- CK were incubated i; 1. 75 mL Buffer B at 30 C for 2-A minutes. The reaction was initiated by adding from 6 uB to 195 uB ADP. The creatine concentrations were 0 mB (a), 5 mB (o), 20 mB (A), and 50 mB (A). The observed maximum velocity corresponds to 320 nmole 0 /min/nmole cytochrome 333. 2 b. Plot of the Km value versus the concentration of creatine for mitochondria (various symbols) or mitoplasts (D). Rates of creatine phosphate synthesis with mitochondria and mitoplasts. Washed mitochondria (o, 0.18 nmole cytochrome aa 0. A IU MiMi- CK), mitoplasts (A, 0.13 nmole cytochrome a; 0. A IU MiMi- CK), or washed mitoplasts (I, 1. A nmole cytoéhrome aa , 0.A IU MiMi-CK) were incubated in 1.5 mL Buffer B at 30 C for 1-2 minutes in the presence of 5 mB 8- hydroxybutyrate. The reaction was started by adding a 1:1 mixture of magnesium acetate and ATP which corresponds to 170 uB total ATP. The open symbols are the ATP concentrations, the closed symbols are the creatine phosphate concentrations. The respiration of washed mitoplasts was inhibited to 30% of normal by adding 3.1 mole carboxyatractlyoside per mole cytochrome 23 The values are the mean of two determinations. The fiates of creatine phosphate synthesis are theoretical lines representing the calculated creatine phosphate concentrations using a fourth order Runge-Kutta numerical integration of Equation VI-A. 180 V 331 I 1 IN ’ A (nmole/thin) 1.0571 7521 - o" atine 150 32:) ' O ' ‘ 0.05 515 0 20 40 60 [Creatine] (my) ,/;;’3/ ‘4’:”’/:1/// 1 -0.2 -0.1 0 0.1 I/[ADP] (1194)" [U IU t 200 . u :1 1 pM CrPi or ATP I00 ‘ time (min) 181 to calculate the [ADP]SS concentration using the measured values for this system (see legend to Table VI-3) and the apparent Km values for ATP which are listed in Table VI-3. Equation VI-A predicts values of [ADP]SS which are close to those measured directly. These data support the observations made earlier that the OMM is the responsible for the preferential coupling of MiMi-CK in chicken heart mitochondria. Discussion The chicken heart mitochondria:MiMi-CK system appears similar to the rabbit heart (10, 11) and rat heart (5 - 7, 12 - 1A) systems in that MiMi-CK and the translocase are preferentially coupled. We define preferential coupling as a decreased Km of an enzyme for a substrate, when measured in the presence of another coupling enzyme, versus the Km value determined by a direct measurement of the enzyme activity. When Ka and Ra values for MiMi-CK are compared, the values are 3 fold lower when determined by coupling the reaction to oxidative phosphorylation as compared to the values measured for free enzyme and enzyme bound to mitoplasts (Table VI-1). While these differences are small, they do accurately predict the steady state concentrations of ATP and ADP measured directly (Table VI-2 and VI-3). The kinetic differences also translate into different rates of creatine phosphate synthesis when the bulk steady state ATP concentration is maintained near 1A0 uB (Figure VI-A); the rate of creatine phosphate production for intact mitochondria is twice that of the free enzyme and approximately 1.7 times that of the enzyme bound to the IMM. Although the [ADP]SS values in Table VI-3 are higher for the bound enzyme (experiment b), this is a result of different V2 values for the two experiments (different rates 182 of oxidative phosphorylation). Extrapolating these results to the case where both V1 and V2 values are identical shows that the effect of lower Km values for translocase and MiMi-CK is to increase the steady state solution concentration of ADP; the steady state concentration of ADP increases two fold when the system is preferentially coupled. These results show that the kinetic constants measured under the various conditions can be used as practical kinetic constants for MiMi- CK to predict concentrations of the relevant substrates in the bulk solution. The lack of an effect of the OMM on the Kb and Kb values for creatine suggests that the outer membrane probably stabilizes an unstirred layer which exists in between the IMM and the OMM apparently by creating a partial diffusion barrier to charged species only. This result has been previously observed for ATP (11), ADP (2A) and for individual ion species such as magnesium and inorganic phosphate (25, 26). The diffusion barrier results in an increased concentration of ATP and ADP in the inter membrane space versus the bulk concentration when preferential coupling occurs. The magnitude of the concentration increase can be estimated by assuming that the Km values are true dissociation constants. If the true Km value is that measured in the absence of a diffusion barrier and the observed Km value is that measured in the presence of a diffusion barrier, then the ratio of the concentrations of substrate outside the barrier to inside the barrier (So/Si) is defined by Equation VI-9. [VI-9] S /S = real K /observed K o i m m 183 For the MiMi-CKztranslocase system, we can calculate that the MgATP-2 concentration is 3.3 fold higher in the presence of respiration (see Figure VI-3) and the ADP concentration is two fold higher in the presence of creatine (see Figure VI-A). The preferential coupling seen in the above experiments is absent when the outer mitochondrial membrane is removed by digitonin treatment, even though about 70% of the MiMi-CK is associated with the IMM. These results are different from those of Saks £2.2l' (6) who observed preferential coupling in the absence of the outer membrane. However, if we assume that the rat heart system is similar to the chicken heart system in binding characteristics, only 10% of the MiMi- CK is bound to the IMM under the conditions of Saks §£.§l° (see Chapter V). Furthermore, as rat heart mitochondria contain approximately 9 times more MiMi-CK than chicken heart mitochondria (M. DeLuca, personal communication), the rate limiting step in the experiments of Saks 33 El° (6) when MiMi-CK is coupled to the translocase, should not be MiMi- CK but oxidative phosphorylation. Comparing the MiMi-CK activity in the experiments of Saks 22 El. (6) to the amount of enzyme previously reported for this system (27), shows that they have approximately 10% of the activity which should be present. Lastly, if we interpret the results of their experiments in terms of the the direct binding of MiMi-CK to nucleotide translocase, it is the activities of the individual proteins, and not the overall (solution) activities which are important in determining if preferential coupling occurs. They must directly demonstrate that the turnover number of the nucleotide translocase is greater than that of MiMi-CK so that the rate limiting 18A step is not the nucleotide translocase but MiMi-CK in their experiments. References 5. 10. 11. 12. Bessman, S.P., and Geiger, P. (1981) Science 211, AA8-A53. Bessman, S.P., and Carpenter, C.L. (1985) Ann. Rev. Biochem. 5A, 831-862. Jacobus, W.E., and Lehninger, A.L. (1973) B. Biol. Chem. 2A8, u803-u81o. Jacobs, H., Heldt, H.W., and Klingenberg, M. (1973) Biochem. Biophys. Res. Comm. 16, 516-521. Saks, V.A., Kupriyanov, V.V., Elizarova, G.V., and Jacobus, W.E. (1980) 3. Biol. Chem. 255, 755-763. Saks, V.A., Kuznetsov, A.V., Kupriyanov, V.V., Miceli, M.V., and Jacobus, W.E. (1985) 3. Biol. Chem. 260, 7757-776A. Jacobus, W.E., and Saks, V.A., (1982) Arch. Biochem. Biophys. 219, 167-178. Moreadith, R.W., and Jacobus, W.E. (1982) 3. Biol. Chem. 257. 899-905. Barbour, R.L., Ribaudo, J., and Chan, S.H.P. (198A) B. Blgl. 9229' 259, 82A6-8251. Erickson-Viitanen, S., Viitanen, P., Geiger, P.J., Yang, W.C.T., and Bessman, S.P. (1982) B. Biol. Chem. 257. 1A395‘ 1AAOA. Erickson-Viitanen, S., Geiger, P., Viitanen, P., and Bessman S.P., (1982) 3. Biol. Chem. 257, 1AA05-1AA11. Saks, V.A., Lipina, N.V., Smirnov, V.N., and Chazov, E.I. 13. 1A. 15. 16. 17. 18. 19. 20. 21. 22. 23. 2A. 25. 26. 27. 185 (1976) Arch. Biochem. Biophys. 173. 3A-A1. Jacobus, W.E. (1985) Ann. Rev. Physiol. A7, 707-725. Vandegaer, K.M., and Jacobus, W.E. (1982) Biochem. Biophys. Egg. 92B 109, AA2-AA8. Hall, N., and DeLuca, M. (198A) Arch. Biochem. Biophys. 229, A77-A82. Bennett, V.D., Hall, N., DeLuca, M., and Suelter, C.H. (1985) Arch. Biochem. Biophys. 2A0, 380-391. Wenger, W.C., Murphy, M.P., Brierley, G.P., and Altschuld, R.A. (1985) B. Bioenegg. Biomemb. 17, 295-303. Toth, P.P., Ferguson-Miller, S., and Suelter, C.H. (1986) Meth. Enzymol. 126, Strehler, B.L. (1963) Meth. Enzymtic Anal. 2, 563-56A. Jacobus, W.E., Moreadith, R. W., and Vandegaer, K.M. (1982) _An_n. g. 1. 593g. £1. Inn, 73-89. Storer, A., and Cornish-Bowden, A. (1976) Biochem. B. 159, 1'5. Wilkinson, G.N. (1963) Biochem. B. 80, 32A-332. Vandegaer, K.M., and Jacobus, W.E. (1982) Biochem. Biophys. Egg. Comm. 109, AA2-AA8. Brdzicka, D. (1978) Hoppe-Seyler's B. Physiol. Chem. 359, 1063. Brierley, G., and O'Brien, R.L. (1965) B. Biol. Chem. 2A0, “532-A539. Pfaff, E., Klingenberg, M., Ritt, E., and Vogell, W. Eur. B. Biochem. 5, 222-232. Kuznetov, A.V., and Saks, V.A. (1986) Biochem. Biophys. Res. Comm. 13A. 359“366. Chapter VII GENERAL CONCLUSION Chicken heart MiMi-CK binds to the outside of the inner mitochondrial membrane (IMM). Increasing ionic strength causes the release of enzyme. Only 50% is released from intact mitochondria while 95% is released from mitoplasts. Titrating mitoplasts with homogeneous MiMi-CK results in the binding of 1 mole of MiMi-CK per 3 moles of nucleotide translocase with a KD = 200 nB at pH 7.A. The binding is also pH dependent; increasing pH decreases the affinity. Measuring the KD values as a function of pH indicates that a group(s) with a pKa of about 6 must be protonated for binding to occur. The binding of MiMi-CK on the outside of the inner membrane results in a preferential access of MiMi-CK for ATP released by the translocase and of translocase for ADP generated by MiMi-CK. The preferential coupling of these two enzymes is shown by a three fold lower Ka and Ea value for MgATP.2 when the reaction is coupled to oxidative phosphorylation. These lower values translate into higher ADP and lower ATP steady state concentrations in the presence of creatine showing that the measured Ka values reflect the solution kinetics of the system: lower apparent Ka values result in higher activity and higher steady state ADP levels. Thus the translocase and MiMi-CK enzymes behave as two coupled enzymes with different solution Km values under different conditions. When coupled to oxidative phosphorylation, creatine phosphate synthesis is greater at identical 186 187 solution ATP concentrations and the translocase activity is higher at identical solution ADP concentrations. The outer mitochondrial membrane is responsible for the observed lowering of the Ka and Ra values: digitonin prepared mitoplasts show a much smaller preferential coupling even though more than 70% of the enzyme is bound to the IMM under the conditions of the experiment. This latter suggests that the chicken heart outer mitochondrial membrane stabilizes a diffusional barrier to ATP which becomes the rate limiting step when solution ATP is used to measure the kinetic constants of MiMi-CK. This conclusion is strengthened by the measurement of a Ka value for MgATP--2 of 800 uB using intact mitochondria in the presence of oligomycin A as compared to 125 uB for soluble homogeneous MiMi-CK. APPENDICIES all 10 20 30 A0 50 60 70 80 90 100 110 120 130 1A0 150 160 170 180 190 200 210 220 230 2A0 250 260 270 280 290 300 310 320 330 3A0 350 360 370 380 390 A00 A10 A20 A30 AAO Appendix A This is the program listing for WILMANA, written in basic, which ows the calculation of kinetic parameters from initial velocity data. REM WILMANA: Written by S. P. J. Brooks LG - 2. 302585093# ’ PP$-" press any key to continue" PPP$-"press 'c' to continue or 'p' to print" PPPP$-"Is the printer turned on ?" PPPPP$-" " 55$.nxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxn CLS:LOCATE 1,1,0:WIDTH A0:COLOR 15.A:LOCATE 10,13:PRINT" " LOCATE 11, 13: PRINT" WILMAN A " :LOCATE 12,13,1:PRINT" " LOCATE 19,1 :PRINT"copyright 1985, M. S. U. " LOCATE 21, 1, O: PRINT PP$ ANYKEY$= INKEY$: IF ANYKEY$="" THEN 120 DIM S(A0), VEL(A0), RESID(A0), RVM(1600), KBV(1600) ,,W(A0) LPY$(75, A5), A$(A5). PT(A0) COLOR 7, 0: CLS: LOCATE 1,1,0:PRINT:PRINT:PRINT"This program calculates the Michaelis" PRINT:PRINT"constant (Km) and Vmax from substrate" PRINT:PRINT"concentrations and initial velocities" PRINT:PRINT"according to one of four different" PRINT:PRINT"estimation methods." PRINT:PRINT:PRINT"This program also tests for the" PRINT:PRINT"presence of outliers based on criteria" PRINT: PRINT"suggested by B. Mannervik, Meth." PRINT. PRINT"Enzymol., 87, 370-390 (1982). " PRINT: PRINT. PRINT PP$¢ ENT$=INKEY$= IF ENT$="" THEN 2A0:ZA=0:N1=0 REMxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx REM REM MENU REM REMNNxENxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx CLS:LOCATE 2,1,0:COLOR 15,A:PRINT"You may do any of the following:" COLOR 7,0:PRINT:PRINT" ‘ 1. Input new data pairs" PRINT:PRINT" 2. Add data to existing data" PRINT:PRINT" 3. Edit and/or review the data" PRINT:PRINT" A. Calculate Km and Vmax;" PRINT" plot'data and residuals." PRINT" (with and without outliers)" PRINT:PRINT" H. Help menu" PRINT:PRINT" E. Exit the program" ANS$=INKEY$: IF ANS$="" THEN 390 IF ANS$="e" OR ANS$="E" THEN 6230 IF ANS$="h" 0R ANS$="H" THEN 5560 A1- VAL(ANS$): IF A1<1 OR A1) A OR A1<>INT(A1) THEN 390 IF N1<1 AND A1-1 OR N1>.1 THEN A70 LOCATE 20,1,0:COLOR 15,A:PRINT"You must enter data PRIOR ":PRINT"to selection of this option":COLOR 7,0:PRINT:PRINT PP$ A50 A60 A70 A80 A90 500 510 520 530 5A0 550 560 570 580 590 600 610 620 630 6A0 650 660 670 680 690 700 710 720 730 7A0 750 760 770 780 790 800 810 820 830 8A0 850 860 870 880 Appendix A (continued) AKEY$=INKEY$:IF AKEY$ - "" THEN A50 LOCATE 20,1,0:PRINT" ":PRINT" ":PRINT:PRINT" ":GOTO 390 ON A1 GOTO A80,A80,790,2120 REMxxxxxxxNxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx REM REM option #1: Enter data pairs REM option #2: Add data to existing data REM REM******§N*******************************************§**** IF A1-2 THEN 620 IF N1<1 THEN 660 CLS:LOCATE 11,5:COLOR 15,A:PRINT"NOTE: This will erase all" LOCATE 12,5:PRINT"previously entered data 1!" PRINT:PRINT:PRINT"Press 'r' to return to the menu" PRINT"Press 'c' to continue" ANS$-INKEY$:IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"r" AND ANS$<>"R" THEN 600 COLOR 7,0:IF ANS$-"r" 0R ANS$="R" THEN 250 ELSE 660 IF N1>=A0 THEN CLS:COLOR 15,A:LOCATE 11,A:PRINT"A0 data pairs is the limit !!":COLOR 7,0:PRINT:PRINT:GOTO 760 ES-N1+1:IF N1<12 THEN EN-13:GOTO 670 IF N1<27 THEN EN=27:GOTO 670 IF N1<-A0 THEN EN=A0:GOTO 670 EN-13:ES=1:LI=0 IF A1-2 THEN LI=N1 CLS:COLOR 15,1:PRINT"press '*' after entering last data pair" LOCATE 3,1:PRINT" pair # ":LOCATE 3,1A:PRINT" [S] ":LOCATE 3,26: PRINT" v " FOR I- ES TO EN: COLOR 15,11LOCATE I+5-LI, 1: PRINT I: COLOR 7, O: LOCATE I+5-LI, 1A: INPUT S$ IF S$-"*" THEN 750 ELSE S(I)-VAL(S$) LOCATE I+5-LI,26:INPUT VEL(I):NEXT I IF I-1A THEN EN=27:ES=1A:LI=13:GOTO 680 IF I-28 THEN ENaAO: ES= 28: LI-27: GOTO 680 N1-I- 1: COLOR 15, 1 PRINT. PRINT N1"data pairs have been entered": GOSUB 5A30 COLOR 7, 0: PRINT:PRINT PP$=ZA=0 ANS$=INKEY$:IF ANS$-"" THEN 770 GOTO 250 REM*************************§*****I“!*********************** REM REM OPTION #3: Review the data REM OPTION #5: List data in plot format REM OPTION #7: List residuals REM REM***********NNNNN**************************§**N**&******* IF A1<>5 THEN 930 CLS:LOCATE A,1,0:COLOR 15,1:PRINT"In what format do you want the data ?":COLOR 7,0 LOCATE 7,1:PRINT"(a) v versus [8]" 890 900 910 920 930 9A0 950 960 970 980 990 1000 1010 1020 1030 10A0 1050 1060 1070 1080 1090 1100 1110 1120 1130 11A0 1150 1160 1170 1180 1190 1200 1210 1220 1230 12A0 1250 1260 1270 1280 1290 1300 1310 1320 Appendix A (continued) PRINT:PRINT"(b) 1/v versus 1/[S]" PRINT:PRINT"(c) v/[S] versus v" PRINT:PRINT"(d) v/Vmax versus log([SJIKm)" 0A$=INKEY$=IF (OA$<"a" OR OA$>"d") AND (OA$<"A" OR OA$>"D") THEN 920 IF N1<-13 THEN EN-N1:ES=1:LI-0:GOT0 950 EN-13:ES-1:LI-O CLS:LOCATE 1,1,0:COLOR 15,1:IF A1=3 THEN PRINT:PRINT" PAIR #": LOCATE 2,13:PRINT" [S] ":LOCATE 2,26:PRINT" v " IF A1<>5 THEN 1020 PRINT:PRINT" pair # ":LOCATE 2,12 IF 0A$-"a" 0R 0A$-"A" THEN PRINT" [S] ":LOCATE 2,25:PRINT" v " IF OA$-"b" 0R OA$-"B" THEN PRINT" 1/[S]":LOCATE 2,25:PRINT" 1/v " IF OA$-"c" 0R OA$-"C" THEN PRINT" v ":LOCATE 2,25:PRINT" v/[S] " IF OA$-"d" 0R 0A$-"D" THEN PRINT" log([S]/Km) ":LOCATE 2,25: PRINT" v/Vmax " IF A1-7 AND PN>0 THEN COLOR 15,A:PRINT"Pair #";PN;"removed from S.D. calculation":COLOR 15,1' IF A1-7 THEN LOCATE 2,1 :PRINT" # " :LOCATE 2, 8. PRINT" resid. " :LOCATE 2, 22: PRINT" 1 28. D. " :LOCATE 2, 32: PRINT" >28. D 2 n. PRINT: FOR I-ES TO EN ‘ COLOR 15,1 :LOCATE 3+I- LI, 2: IF A1-3 0R A1-5 THEN PRINT I, COLOR 7, 0: IF A1-3 THEN PRINT TAB(12) S(I) TAB(25) VEL(I): GOTO 1170 IF A1<>5 THEN 1130 IF OA$-"a" OR OA$-"A" THEN STEMP - S(I):VTEMP - VEL(I) IF OA$-"b" 0R.OA$-"B" THEN STEMP - 1/S(I):VTEMP - 1/VEL(I) IF OA$-"c" 0R OA$-"C" THEN STEMP - VEL(I):VTEMP - VEL(I)/S(I) IF 0A$-"d" OR OA$-"D" THEN STEMP - LOG(S(I)/KM)/LG:VTEMP - VEL(I)/VM PRINT TAB(10) STEMP TAB(2A) VTEMP:GOTO 1170 PC:INT(RESID(I)/(SQR(SE)*2)*1000)/10 COLOR 15,1:LOCATE 3+I-LI,1:PRINT I; COLOR 7,0:PRINT TAB(6) RESID(I) TAB(23) PC:IF ABS(PC)<100 THEN 1170 COLOR 15,A:LOCATE 3+I-LI.35:PRINT" * ":COLOR 7,0 NEXT I: IF A1-3 THEN 1A3O REM****i*******l*********§********************************* REM REM continue listing the data REM REM¥§§§§§§§§§§§§§§§§§§§§**************X1******************** IF N1-I-1 THEN 13A0 LOCATE A+I~LI:PRINT PP$ ANS$-INKEY$:IF ANS$-"" THEN 1250 IF N1<-27 AND I<28 THEN EN'N1HE3813 LI-12: GOTO 950 IF N1>27 AND I<28 THEN EN-27: ES-13: LI-12: GOTO 950 EN‘N1HES‘27 LI-26: GOTO 950 REM*****!***§*§****************§****************§********** REM REM stOp listing the data REM 1330 1340 1350 1360 1370 1380 1390 1400 1410 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 1540 1550 1560 1570 1580 1590 1600 1610 1620 1630 1640 1650 1660 1670 1680 1690 1700 1710 1720 1730 1740 1750 1760 1770 1780 Appendix A (continued) REM*§***************§************************************** IF A1-7 AND N1+1-I THEN COLOR 15,1 HPRINT PRINT: PRINT"sequence = " “88$ PRINT AA$= COLOR 7. 0 IF (A1-7 AND AAA1 -0) OR A1-5 THEN 1390 PRINT: PRINT PP$ ANS$-INKEY$: IF ANS$-"" THEN 1370 CLS: GOTO 2230 PRINT. PRINT PPP$ ANS$=INKEY$:IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"p" AND ANS$<>"P" THEN 1400 IF ANS$="p " OR ANS$-"P" THEN 1750 COLOR 7, 0: pGOTO 3650 REM**************§************************************§**** REM REM continue with option #3 REM REM***************iii!*****************i*****************N! LOCATE 5+I-LI,1:PRINT"Press 'e' to edit data" PRINT"Press 'd' to delete data" PRINT"Press 'c' to continue" ANS$=INKEY$: IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"e" AND ANS$<>"E" AND ANS$<>"d" AND ANS$<> "D" THEN 1510 IF ANS$<>"c" AND ANS$<> "C" THEN 1570 ‘ IF I-N1+1 THEN COLOR 7,0:WIDTH 40:GOTO 250 IF I-14 AND N1<27 THEN EN-N1 HES'13 LI-12: GOTO 950 IF I-14 AND N1>27 THEN EN-27: ES-13: LI-12: GOTO 950 EN=N1HESIZ7 LI-26: GOTO 950 ' Z4=O: LOCATE I+5-LI: COLOR 15,4:PRINT"Hhich data pair do you want to" COLOR 7,0 PRINT" -- " PRINT" " LOCATE 6+I-LI,1:COLOR 15,4 IF ANS$<>"e" AND ANS$<>"E" THEN 1690 INPUT"ed1t ";N2 IF N2EN THEN 1610 LOCATE 3+N2-LI,2:PRINT N2‘ LOCATE 3+N2'LI,13:INPUT" ";S(N2) LOCATE 3+N2-LI,25:INPUT" ":VEL(N2) COLOR 7,0:GOSUB 5430: GOTO 790 INPUT"de1ete ":N2:COLOR 7,0 IF N2<1 OR N2>N1 THEN 1610 N1-N1-1:FOR I-N2 TO N1:VEL(I)-VEL(I+1):S(I)-S(I+1):NEXT I: GOTO 790 REM********************§*!*********§!*****§******************* REM REM print results to line printer REM REMNNN**********&!**§***********************l****************X IF A1-7 THEN LOCF=8+I~LI ELSE LOCF-4+I-LI LOCATE LOCF:PRINT PPPPP$:COLOR 15,A:LOCATE LOCFzPRINT PPPP$:COLOR 1790 1800 1810 1820 1830 18u0 1850 1860 1870 1880 1890 1900 1910 1920 1930 19H0 1950 1960 1970 1980 1990 2000 2010 2020 2030 2030 2050 2060 2070 2080 2090 2100 2110 2120 2130 21uo 2150 2160 2170 2180 2190 2200 Appendix A (continued) 7.0 PRINT. PRINT PP$ ANS$-INKEY$: IF ANS$-""THEN 1800 LPRINT: LPRINT EE$ IF A1-7 THEN 1950 LPRINT:LPRINT"Data listing:":LPRINT"- " LPRINT:LPRINT"pair #" TAB(10) "[substrateJ" TAB(27) "velocity"; IF ou$-"a" 0R ou$-"A" THEN LPRINT TAB(uu) " " IF ou$-"b" 0R ou$-"B" THEN LPRINT TAB(uu) "1/[substrate]" TAB(61) "1/velocity" ‘ ' IF ou$-"c" 0R ou$="c" THEN LPRINT TAB(uu) "velocity/[substrate]" IF ou$-"d" on ou$-"o" THEN LPRINT TAB(uu) ”log([sub.]/Km)" TAB(61) "velocity/Vmax" ‘ LPRINT" ------ " TAB(10) " ----------- " TAB(27) " -------- "; IF ou$-"a" on ou$-"A" THEN LPRINT TAB(uu) " " IF Gus-"b" 0R ou$-"B" THEN LPRINT TAB(uu) " ---------- " TAB(61) "—- IF 0u$="c" 0R 0M$-"C" THEN LPRINT TAB(NM) "---e ---------------- " IF 0u$-"d" 0R 0H$-"D" THEN LPRINT TAB(NM) " -------------- " TAB(61) " ———- n GOTO 1980 LPRINT:LPRINT:LPRINT"Results of WILMANM calculations: ":CC$ LPRINT:LPRINT:LPRINT"pair #" TAB(17) "[substrateJ" TAB(3M) "velocity" TAB(51) "residual" TAB(68) "% of 2 S.D." LPRINT" ------ " TAB(17) " " TAB(3N) "-*-*----" TAB(51) "- ....... '1 TAB(68) fl ' -— fl ' LPRINT: FOR I-1 T0 N1 IF A1-7 THEN PC-INT(RESID(I)/(SQR(SE)*2)*1000)/10:00T0 2060 LPRINT I TAB(10) 8(1) TAB(27) VEL(I); IF 0h$-"a" OR‘OU$-"A" THEN LPRINT TAB(NM) " " IF 0u$-"b" 0R 0u$-"B" THEN LPRINT TAB(HH) 1/S(I) TAB(61)1/VEL(I) IF 0H$-"c" 0R 0u$-"C" THEN LPRINT TAB(HM) VEL(I)/S(I) IF 0u$-"d" 0R 0N$-"D" THEN LPRINT TAB(HN) L00(S(I)/KM)/LG TAB(61) VEL(I)/VM GOTO 2070 LPRINT I TAB(17) S(I) TAB(3H) VEL(I) TAB(51) RESID(I) TAB(68) PC NEXT I:IF A1-5 THEN 2100 IF PN>0 THEN LPRINT:LPRINT"Pair #";PN;"removed from analysis" LPRINT:LPRINT"sequence of residuals (in order of increasing velocity) - ";BB$:LPRINT AA$ LPRINT:LPRINT EE$ GOTO 3650 REM¥**********************ii!!!******************************* REM REM Option #H: Km and Vmax estimation --> leads to sub-menu REM REM***ii*i!*§**§§***i*******l************************§******** CLS:LOCATE 1,1,0 PN=0:IF zuao THEN 2310 CLS:LOCATE 6,1:COLOR‘15,u:PRINT"Do you wish to eliminate outliers" PRINT"(y-yes,'n-no) ?":COLOR 7,0 2210 2220 2230 2240 2250 2260 2270 2280 2290 2300 2310 2320 2330 2340 2350 2360 2370 2380 2390 2400 2410 2420 2430 2440 2450 2460 2470 2480 2490 2500 2510 2520 2530 2540 2550 2560 2570 2580 2590 2600 2610 2620 2630 2640 2650 2660 2670 2680 Appendi x A (continued) ANS4$=INKEY$z IF ANS4$<>"y" AND ANS4$<>"Y"AND ANS4$<>"n" AND ANS4$<> "N" THEN 221 0 IF ANS4$-"n" OR ANS4$="N" THEN 22=2: PN= o. GOTO 2310 AAA1-O: COLOR 15,1:LOCATE 10,1 :PRINT"Enter outlier pair #"z COLOR 7, 0 COLOR 15.3:PRINT:PRINT:PRINT"(T0 display residuals and possible" PRINT"outliers enter -1)" PRINT:PRINT"(TO calculate Km and Vmax without" PRINT"removing an outlier enter 0)":COLOR 7,0 LOCATE 10,23:COLOR 15,1:INPUT PN:COLOR 7,0 IF PN<-1 OR PN>N1 THEN LOCATE 10,23:PRINT" IF PN - -1 THEN A1=7zAAA1-1:GOTO 4390 CLS:PRINT:COLOR 15,2:PRINT"Parameter estimation method desired ?" COLOR 7,0:PRINT:PRINT"(a) Linear regression (Wilkinson)" PRINT" (Biochem J., 1961, 80, 324)":PRINT" "; COLOR 15,1:PRINT"(w i Vmax°2*v“2/(Km+[8])“2))":COLOR 7,0 PRINT:PRINT"(b) Linear regression (C.-Bowden)*”:PRINT" "; COLOR 15,1:PRINT"(w - Vmax*v“2/(Km+[S])[S])":COLOR 7,0 PRINT:PRINT"(c) Linear regression (J. & Lumry)" PRINT" (C. R. Trav. Lab. Carls., 1961,)" PRINT" 32, 185) ‘"; COLOR 15,1:PRINT"(W - V22/[S]“2)": COLOR 7. O PRINT. PRINT"(d) Non-parameter estimate (C. -Bowden)*" PRINT" "; COLOR 15,4. PRINT"N0te 1!"; COLOR 7,0:PRINT" Different velocity measure-" ":GOTO 2280 PRINT" ments at a single substrate cone." PRINT" are not allowed with this option 1!" PRINT: PRINT. PRINT"* (Principals of Enzyme Kinetics, (1976)" PRINT" Butterworths Pub. Co. )" F$-INKEY$: IF (F$<"A" OR F$>"D") AND (F$<"a" OR F$>"d") THEN 2490 IF F$-"a" 0R F$-"A" THEN CC$-"Linear regression (Wilkinson)" IF F$-"b" 0R F$-"B" THEN CC$-"Linear regression (Intermed.)" IF F$-"c" 0R F$-"C" THEN CC$="Linear regression (J. & Lumry)" IF F$-"d" 0R F$-"D" THEN CC$="Non-parameter (C.-Bowden)":GOT0 2810 REM*****§****I‘I******************************** REM REM Linear regression estimation REM REM******************************************** CLS:K0-O:N3-0 E1-O:E2=O:E3-O:E4=O:E5=O FOR I-1 T0 N1 IF I-PN THEN 2730 IF N3-0 AND (F$-"a" IF N3-0 AND (F$-"b" OR F$="A") THEN W(I)-VEL(I)“4/S(I)“2 OR F$-"B") THEN H(I)-VEL(I)“3/S(I)“2 IF N3>O AND (F$-"a" OR F$-"A") THEN W(I)=VEL(I)“2*VM"2/(KM+S(1))“2 IF N3>0 AND (F$-"b" OR F$-"B") THEN W(I)-VEL(I)‘2*VM/((KM+S(I))*S(I)) IF F$-"c" OR F$-"C" THEN W(I) - VEL(I)“2/S(I)‘2 E1-E1+W(I) 2690 2700 2710 2720 2730 2740 2750 2760 2770 2780 2790 2800 2810 2820 2830 2840 2850 2860 2870 2880 2890 2900 2910 2920 2930 2940 2950 2960 2970 2980 2990 3000 3010 3020 3030 3040 3050 3060 3070 3080 3090 3100 3110 3120 3130 3140 3150 3160 3170 3180 3190 Appendix A (continued) E2-E2+H(I)*S(I)‘2 EB-E3+W(I)*S(I) E4-E4+N(I)*S(I)‘2/VEL(I) E5-E5+H(I)*S(I)/VEL(I) NEXT I KM-(E5*E2-E3*E4)/(E1*E4-E3*E5) VM-(E1*E2-E3‘2)/(E1*E4-E3*E5) IF F$-"c" OR F$-"C" THEN 3110 KThABS(INT((KM-K0)/KM*10000)): IF KT<1 THEN DD$="": GOTO 3110 K0=KM: N3-N3+1 IF N3<10 THEN 2600 DD$-"Not converged by 10th iteration !!":GOTO 3110 REM******************************************** REM REM Non-parameter estimation REM REM§¥¥§§§§§¥§§§§§***************************ii! CLS:DD$-"":K=O: FOR I-1 T0 N1: IF I-PN THEN 2920 FOR J-1 T0 N1: IF JsPN THEN 2910 IF I-J THEN 2910 K-K+1:RVM(K)-(S(J)/VEL(J) S(I)/VEL(I))/(S(J)-S(I)) KBV(K)= S(I)*(1/VEL(I)- RVM(K)) NEXT J NEXT I:GAP =-INT(K/2) FLO-1:FOR I=1 T0 K-GAP: IF RVM(I)<=RVM(I+GAP) THEN 2950 TEMP-RVM(I):RVM(I)-RVM(I+GAP):RVM(I+GAP)=TEMP:FLG=0 NEXT I:IF FLGaO THEN 2930 GAP-INT(GAP/Z):IF GAP>O THEN 2930 IF K/2- INT(K/2) THEN I1=K/2: VM=. 5*(1/RVM(I1)+1/RVM(1+I1)):J1=2 IF K/2<>INT(K/2) THEN I1-INT(K/2)+1:VM=1/(RVM(I1)):J1-1 GAP-INT(K/Z) FLG-1:FOR I=1 TO K-GAleF KBV(I)<=KBV(I+GAP) THEN 3020 TEMPéKBV(I):KBV(I)=KBV(I+GAP):KBV(I+OAP)-TEMP:FLG=O NEXT I:IF FLO-0 THEN 3000 GAP-INT(GAP/Z):IF GAP>0 THEN 3000 IF J1-1 THEN KM=VM*KBV(I1) IF J1-2 THEN KM=VM*(KBV(I1)+KBV(I1+1))/2 REM§§§§§§§§¥§§§§******************************* REM REM Print results to the screen REM REM**************************************1!!-X-*** CLS:PRINT:PRINT"Results: "; CC$ PRINT:COLOR 15,4:PRINT DD$:COLOR 7,0 SS-O IF F$-"d" OR F$="D" THEN E1=O=E2=O:E3=O FOR I-1 T0 N1:IF IaPN THEN 3220 IF F$<>"d" AND F$<>"D" THEN 3190 H(I) - VEL(I)‘2*VM/((KM + S(I))*S(I)) E1-E1+W(I):E2=E2+W(I)*S(I)“2:E3=E3+W(I)*S(I) ERRORF - S(I)/VEL(I) - S(I)/VM - KM/VM 3200 3210 3220 3230 3240 3250 3260 3270 3280 3290 3300 3310 3320 3330 33u0 3350 3360 3370 3380 3390 3u00 3110 3u20 3430 3uuo 3450 3460 3470 3480 3490 3500 3510 3520 3530 3540 3550 3560 3570 3580 3590 3600 3610 3620 3630 Appendix A (continued) RESID(I) - -ERRORF*SQR(W(I)) SS - SS + W(I)*ERRORF“2 NEXT I:IF PN=0 THEN 3270 IF F$-"a" OR F$-"A" THEN H(PN)-VEL(PN)“2*VM‘2/(KM+S(PN))‘2 IF F$-"b" OR F$-"B" OR F$-"d" 0R F$-"D" THEN W(PN)-VEL(PN)‘2*VM/((KM+S(PN))*S(PN)) IF F$-"c" OR F$-"C" THEN N(PN)-VEL(PN)‘2/S(PN)‘2 RESID(PN)- (S(PN)/VM+KM/VM-S(PN)/VEL(PN))*SQR(W(PN)) BT - E1*E2 - E3‘2 PE=O:IF PN>O THEN PE-1 SE-SS/(N1-2-PE) VK-SQR(VM‘2*SE*(E2+2*KM*E3+KM‘2*E1)/BT) VV-SQR(VM‘4*SE*E1/BT) VS-SQR(SE*E2/BT)' IF PN>0 THEN COLOR 15,4:LOCATE 4,1:PRINT"pair #";PN;"removed from analysis":COLOR 7,0 COLOR 15,1:LOCATE 6,1:PRINT"Km - ";KM LOCATE 8,1:PRINT"Vmax - ";VM LOCATE 10,1:PRINT"Slope (Km/Vmax) - ";KM/VM:COLOR 7,0 COLOR 15,2:LOCATE 13,1:PRINT"std. dev. of Km - ";VK LOCATE 15,1:PRINT"Std. dev. of Vmax - ";VV LOCATE 17,1:PRINT"Std. dev. of slope - ";VS:COLOR 7,0 IF F$-"d" OR F$-"D" THEN COLOR 15,3:LOCATE 19,1:PRINT"** Note: std devs. are approximations !!":COLOR 7,0 IF PN-1 THEN KML1-2 ELSE KML1-1 IF PN=N1 THEN KML2-N1-1 ELSE KML2-N1 KML-S(KML1)/KM*100: KMHhS(KML2)/KM*100 COLOR 15,1:LOCATE 21,1 :PRINT USING "[S] varies from ###. #% to ###. #% of Km"; KML, KMH: COLOR 7, 0 LOCATE 23, 1: PRINT PPP$ Z4-1:ANS$-INKEY$:IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"p" AND ANS$<>"P" THEN 3460 IF ANS$-"c" OR ANS$-"C" THEN 3650 REMNH*§***********************!***iflifliflflifiiflfii**************i REM REM print to line printer REM REM**§*******l§**§***I*§***§*****ii******§**i§**§§§*********N! LOCATE 21,1:PRINT PPPPP$+" ":LOCATE 23,1:PRINT PPPPP$ COLOR 15,4:LOCATE 21,1:PRINT PPPP$:COLOR 7,0:LOCATE 22,1:PRINT PP$ ANS$-INKEY$:IF ANS$-"" THEN 3550 LPRINT:LPRINT EE$ LPRINT:LPRINT"Resu1ts of WILMAN4 calculations: ";CC$ LPRINT:LPRINT "Km - ";KM;" +/- ";VK LPRINT"Vmax . ";VM;" +/- ";VV LPRINT"Km/Vmax - " HKM/VM " +/- "VS LPRINT: LPRINT: LPRINT USING "[S] varies from ###. #1 to ##fi. #1 of Km"; KML, KMH IF F$-"d" OR F$-"D" THEN LPRINT: LPRINT"** Note: std. devs. are approximations !!" IF PN>0 THEN LPRINT:LPRINT"Pair #";PN;"removed from analysis" 3640 3650 3660 3670 3680 3690 3700 3710 3720 3730 3740 3750 3760 3770 3780 3790 3800 3810 3820 3830 3840 3850 3860 3870 3880 3890 3900 3910 3920 3930 3940 3950 3960 3970 3980 3990 4000 4010 4020 4030 4040 4050 4060 4070 Appendix A (continued) LPRINT:LPRINT EE$ REMiiiiiiiiiiiifliiiiiiiii**§****ii!ii!!!***********&********** REM REM Sub-menu REM REMifliiiiifiiiliiiiiiiiflfliiiiiiiii!Niiiiiiifiiiififli**I********Ni CLS: LOCATE 2, 1, O: COLOR 15, 4: PRINT"You may do any of the following:" COLOR 7, O: PRINT:PRINT" 4. Recalculate Km and Vmax" PRINT:PRINT" 5. List data in plot format" PRINT:PRINT" 6; Plot the data" PRINT:PRINT" 7; List residuals" PRINT:PRINT" 8; Plot residuals" PRINT:PRINT" R; Return to main menu" ANS$=INKEY$= IF ANS$-"" THEN 3770 IF ANS$-"r" OR ANS$-"R" THEN 250 A1- VAL(ANS$): IF A1<4 OR A1) 8 OR A1<>INT(A1) THEN 3770 A11-A1-3:ON A11 GOTO 2120,790,3810,4390,4390‘ REM*******i§§§¥§§§§§§§ii§¥i§ii§ifli**§*****ii****************** REM REM OPTION #6: Plot the data REM REM******i!*******i*i***fli*i**§¥**%**§*************I********** CLS: LOCATE 4, 1, 0: COLOR 15,1:PRINT"How do you want to plot the data ?": COLOR 7, 0 ‘ LOCATE 7,1:PRINT"(a) v versus [8]" PRINT:PRINT"(b) 1/v versus 1/[S]" PRINT:PRINT"(c) v/[S] versus v" PRINT:PRINT"(d) v versus logESJ" 06$=INKEY$=IF (06$("a" OR 06$)"d") AND (06$<"A" OR 06$)"D") THEN 3910 TEMP=1=TEMP1-1:IF PN-1 THEN TEMP-2:TEMP1-2 FOR 1:2 TO N1:IF PNaI THEN 3970 IF 06$="a" OR O6$-"A" THEN IF VEL(I)>VEL(TEMP) THEN TEMP=I IF 06$-"b" OR O6$-"B" THEN IF 1/VEL(I)>1/VEL(TEMP) THEN TEMP-I IF 06$="d" OR 06$-"D" THEN IF ‘ ABS(LOG(S(I)/KM)/LG)>ABS(LOG(S(TEMP)/KM)/LG)THEN TEMP=I NEXT I:TEMP2=N1:IF N1-PN THEN TEMP2-N1-1 IF 06$ -"a" 0R 06$-"A" THEN B1-S(TEMP2):BZ-VEL(TEMP) IF 06$ ="b" OR 06$="B" THEN B1-1/S(TEMP1):BZ-1/VEL(TEMP) IF 06$ ="c" OR O6$-"C" THEN BI-VM:BZ-VM/KM IF 06$ ="d" OR O6$-"D" THEN B1=ABS(LOG(S(TEMP)/KM)/LG):BZ=1 CLS:SCREEN 1,0:COLOR 0,1 IF PN>O THEN PRINT"Pair'#";PN;"removed from the graph" IF O6$="a" OR O6$-"A" THEN LOCATE 9,1:PRINT" v":LOCATE 20,28:PRINT"[S]" IF 06$="b" 0R 06$="B" THEN LOCATE 9,1:PRINT" v":LOCATE 20,28:PRINT"1/[S]" 1":PRINT" -":PRINT" IF 06$-"c" 0R 06$-"C" THEN LOCATE 9,1:PRINT" v":PRINT"--- ":PRINT"[S]":LOCATE 20,28:PRINT"v" IF 06$-"d" OR 06$="D" THEN LOCATE 9,1:PRINT" v":PRINT" ---- N080 "100 ”110 "120 “130 “1&0 ”150 “160 N170 N180 N190 N200 N210 N220 N230 N2N0 N250 N260 N270 N280 N290 N300 N310 N320 N330 N3N0 N350 N360 N370 N380 N390 NN00 NN10 NN20 NN30 NNNO NN50 NN60 NN70 NN80 NN90 N500 N510 N520 N530 N5N0 Appendix A (continued) ":PRINT" Vmax":LOCATE 20,28:PRINT"(log[S]/Km)" LOCATE 22,1:PRINT PP$ N090 LINE (30,150)-(300,1N7),2,BE A-33:IF 06$-"d" OR O6$-"D" THEN A-166 LINE (A,10)-((A-3).1N7).2,BE FOR I-1 T0 N1:IE PN - I THEN N180 IE O6$-"a" OR O6$-"A" THEN XCORD=S(I)/B1*267+33:YCORD=1N7- VEL(I)/BZ*137 IF 06$-"b" 0R O6$-"B" THEN XCOHD=(1/S(I))/B1*267+33. YCORD=1N7- (1/VEL(I))/BZ*137 IF 06$-"c" OR 06$-"c" THEN XCORD-VEL(I)/81*257+33:YCORD =1N7- (VEL(I)/S(I))/Bz*127 IF O6$-"d" on 06$-"D" THEN XCORD—33+133. 5*(81+LOG(S(I)/KM)/LG)/B1. YCORD =1N7-VEL(I)/VM*137 CIRCLE (XCORD, YCORD), 2, 1,,,1 NEXT I IF 06$<>"a" AND O6$<>"A" THEN N250 FOR XCORD - 33 T0 300 STEP N XXXI-(XCORD-33)*B1/267:XXX2-(XCORD+N-33)*81/267 YCORDI- 1N7-137*(VM*xxx1/(KM + xxx1))/52 YCORDZ- 1N7-137*(VM*xxx2/(KM + XXX2))/82 LINE (XCORD,YCORD1) - (XCORD+N,y00RDZ).3:NEXT XCORDzGOTO N370 IF 06$<>"d" AND 06$<>"0" THEN N320 FOR XCORD - 33 T0 300 STEP N xxx1-KM*EXP(((x00RD-33)*B1/133.5-31)*L0) xxxz-KM*EXP(((x00R0-29)*B1/133;5-B1)*LG) YCORD1- 1N7-137*(VM*xxx1/(KM +‘XXX1))/VM YCORDZ- 1N7-137*(VM*xxx2/(KM + XXX2))/VM LINE (XCORD,YCORD1) - (XCORD+N,ICOH02),3:NEXT XCORD:GOTO N370 IF O6$-"c" OR 06$~"C" THEN N360 CORD1-1N7—(1/VM)/Bz*137 00E02-1N7-(1/(VM/B1/(KM+1/BI)))/82*137 LINE(33.COR01)-(300,00R02),3:00T0 N370 LINE(33.20)—(290,1N7) ANS$=INKEY$=IF ANS$-"" THEN N370 SCREEN 0,1:COLOR 7,0:GOT0 3650 REM»!MIiiilfliifliflfliii*iiiifiiiiiii“Ht-xi**************§*i****** REM REM Option #7: List residuals REM Option #8: Plot residuals REM REM}!!!*****§************************************************ FOR I-1 T0 N1:PT(I)-I:NEXT I FOR I-1 T0 N1- 1: I1-I: LEAST - VEL(PT(I)) FOR J-1+I T0 N1: IF VEL(PT(J))I THEN TEMP=PT(I): PT(I)-PT(I1). PT(I1)=TEMP NEXT I CLS:M9-0:N9-O:U9-O:A2-O:BB$-"" FOR I-1 T0 N1: IE PT(I)-PN THEN N570 IE RESID(PT(I))0 THEN N9=N9+1:A3=2:BB$=BB$+"+" ”550 "560 N570 N580 N590 N600 N610 N620 N630 N6N0 N650 N660 N670 N680 N690 N700 N710 N720 “730 N7No N750 N760 N770 N780 N796 N800 N810 N820 N830 N8N0 N850 N860 N870 N880 N890 N900 N910 N920 "930 N9No N950 N960 N970 N980 N990 5000 5010 5020 5030 Appendix A (continued) IF RESID(PT(I))-0 THEN BB$-BB$+"0" IF A3<>A2 THEN U9=U9+1 A2-A3:NEXT I IF M9+N9<8 THEN AA$-"Too few points to analyse randomness":GOTO ”750 IF N9INT(K9) THEN K9-(I+1)/2:GOT0 N660 A3-M9-1:AN-K9-1:OOSUB 5300:C1-C9 A3=N9-1:Au-K9-1:GOSUB 5300:CZ=C9 FU - 2*C1*C2:GOTO “710 A3=M9-1:AN=K9-1:GOSUB 5300:C1-C9 A3-N9-1:AN-K9-2:GOSUB 5300:C2-C9 A3=M9-1:Au-K9-2:GOSUB 5300:C3-C9 A3-N9-1:AN-K9-1:GOSUB 5300:CN-C9 FU-C1*C2+03*CN FT-FT+FU:NEXT I A3-N9+M9:Au-M9:GOSUB 5300: FT=FT/C9:Z9=1 IF FT>.OS THEN AA$-"The sequence is random (p>0.95)" IF FT<*.OS THEN AA$-"The sequence is non random (p>O.95)" IF A1-7 THEN 790 ' REM****§*******************N***************§************** REM REM Option #8: Plot residuals (continued) REM REM}!**!**************I***N******!*§********************** TEMP-1:TEMP1-1:IF PN-1 THEN TEMP-2:TEMP1-2 B1-VEL(TEMP):BZ-ABS(RESID(TEMP1)) FOR I-TEMP T0 N1:IF I=PN THEN N860 IF ABS(RESID(I))>BZ THEN BZsABS(RESID(I)) IF VEL(I)>B1 THEN E1-VEL(I) NEXT I:SCREEN 1,0 COLOR 0,1:PRINT"Sequence - ";BB$:PRINT AA$ EN-8:IF PN>O THEN PRINT"Pair #";PN;"removed from graph":EN-7 FOR I-1 T0 EN:PRINT:NEXT I:PRINT"R":PRINT"e":PRINT"S":PRINT"1"; PRINT TAB(28)"Velocity":PRINT"d" LOCATE 23,1:PRINT"Press 'c' to continue or 'p' to plot" LINE (33.30)‘(30.175).2.BF LINE (30,10N)-(300,101),2,BF PN-N1:IF PN-N1 THEN'PN-NI-l FOR I-1 TO N1:IF I=PN THEN H980 XCORD - 33 + VEL(PT(I))/Bl*267 YCORD -102 - (RESID(PT(I))/82)*72:CIRCLE (XCORD,YCORD),2,1,,,1 NEXT I ANS$=INKEY$:IF ANS$<>"C" AND ANS$<>"C" AND ANS$<>"p" AND ANS$<>"P" THEN N990 IF ANS$-"c" 0R ANS$-"C" THEN SCREEN 0,1:COLOR 7,0:GOTO 3650 LOCATE 23,1:PRINT"Is the printer on ? (press any key) " ANS$=INKEY$=IF ANS$-"" THEN 5020 REM************§***************§*********************N******** SOHO 5050 5060 5070 5080 5090 5100 5110 5120 5130 5140 5150 5160 5170 5180 5190 5200 5210 5220 5230 52H0 5250 5260 5270 5280 5290 5300 5310 5320 5330 5340 5350 5360 5370 5380 5390 5N00 5fl10 5N20 5N3O 5NNO 5M50 5N60 5N70 5H80 5390 5500 Appendix A (continued) REM REM plot residuals on line printer REM REM*****§**§****§**************§*§**********************§i§*** CLS:POR I-O T0 N0: A$(I)=" ":NEXT I A$(18)-" r ":A$(17)-" e ":A$(16)=" s ":A$(15)=" i ":A$(13)-" u ":A$(12)=" a ":A$(11)-" l " ' FOR X=O TO 70: FOR Y=O TO 30: LPY$(X, Y)=" ":NEXT Y:NEXT X FOR X-1 T0 70: IE INT(X/10)-X/10 THEN A1$-"|" ELSE A1$-"-" LPI$(X, 15)-A1$: NEXT x * FOR Y-O TO 30: IF INT(Y/3)=Y/3 THEN A1$=" -" ELSE A1$="|" LPI$(0, I)- A1$= NEXT I FOR I-1 T0 N1 :IF I-PN THEN 5180 Y-(RESID(PT(I))+BZ)/BZ*1N. 5: X-VEL(PT(I))/B1*70 LPY$(INT(X+. 5), INT(I+. 5))="0" NEXT I LPRINT: LPRINT: LPRINT EE$: LPRINT: LPRINT"Graph of residuals versus velocity: " :LPRINT" - " HLPRINT LPRINT FOR Y-30 TO 0 STEP -1 LP5$-"": FOR X-O TO 70: LP5$=LP5$+LPY$(X,Y):NEXT x LPRINT " "+A$(I)+LP5$: NEXT I LPRINT: LPRINT" velocity" LPRINT:LPRINT"X interval . ";B1/7:LPRINT"y interval = ";BZ/15*3 LPRINT:LPRINT"Residuals calculated using: ";CC$ LPRINT:LPRINT"sequence of residuals - ";BB$:LPRINT AA$ IF PN>O THEN LPRINT:LPRINT"Pair #";PN;"removed from analysis" LPRINT:LPRINT EE$ SCREEN 0,1:COLOR 7,0: GOTO 3650 REMfifiiiflifiiiifl'lNI-ii‘l!!!ii!-l-*********************************** REM REM Factorial subroutine REM REM*********************************************************** J1-1:J2-1:J3=1:A5-(A3-Au) IF A3<-O THEN A321 IF AN<-O THEN AN-1 IF AS<-O THEN A5=1 FOR I3- A3 TO 1 STEP ~1:J1=J1*I3:NEXT I3 FOR I3 -AN TO 1 STEP -1:J2=J2*I3:NEXT I3 FOR I3-A5 TO 1 STEP -1:J3=J3*I3:NEXT I3 C9-J1/(J2*J3):RETURN REMNN'N'NNNNI'****l****************{11************************ REM REM Sorting subroutine REM . REM*******§********************!********************#11111!!- FOR I3-1 TO N1-1:I1-I3:LE=S(I3) FOR IN=I3+1 TO N1:IF S(IN)"D") AND (ANS$<"a" OR ANS$>"d") THEN 56NO IF ANS$-"d" OR ANS$-"D" THEN 250 IF ANS$<>"a" AND ANS$<>"A" THEN 58NO CLS:LOCATE 2,1,0:COLOR 15,3:PRINT"Removing outliers:":COLOR 7,0 PRINT:PRINT"Outliers are defined as data which are" PRINT"two times greater than the experimental" PRINT"standard deviation. They are indicated" PRINT"by the presence of an '*' in the far" PRINT"right hand column of option #7. Once" PRINT"outliers are identified, return to #N" PRINT"and indicate which value you wish to" PRINT"remove (you can remove only one" PRINT"outlier at a time). The new values" PRINT"of Vmax and Km are calculated without" PRINT"the outlier. Option #7 now shows the" PRINT"new residuals." PRINT:PRINT"See B. Mannervik, 1982, Meth. Enzymol.," PRINT"vol. 87. PP 370-390.":PRINT:PRINT PP$ ' ANS$=INKEI$=IE ANS$="" THEN 5820 GOTO 5560 IE ANS$<>"b" AND ANS$<>"B" THEN 6010 CLS:LOCATE 2,1,0:COLOR 15,3:PRINT"Residual equations:":COLOR 7,0 PRINT:PRINT"The following equation defines the" PRINT"residual shown in option #7 and #8:" PRINT:COLOR 15,1:PRINT"resid - u * d" COLOR 7,0:PRINT:PRINT"where d - -Vmax*v*e/(Km + [8])" PRINT" e - [SJ/v - Km/Vmax - [SJ/Vmax" PRINT"(see menu C for a definition of u)" PRINT: PRINT"One standard deviation (S. D.) is" PRINT"defined as follows: " PRINT:COLOR 15,1:PRINT"S.D. - SQRESUM(u * d‘2)/(n - p)]" COLOR 7,0:PRINT:PRINT"where n is the number of data points" PRINT" p is the number of parameters" PRINT" (for our case p22)" PRINT:PRINT PP$ 5990 6000 6010 6020 6030 GONO 6050 6060 6070 6080 6090 6100 6110 6120 6130 61H0 6150 6160 6170 6180 6190 6200 6210 6220 6230 Appendi x A (continued) ANS$-INKEY$:IF ANS$-"" THEN 5990 GOTO 5560 CLS:LOCATE 2,1,0:COLOR 15.3:PRINT"Km and Vmax estimations:" COLOR 7,0:PRINT:PRINT"Km and Vmax are estimated using" PRINT"either one of the three linear" PRINT"regression procedures or by the" PRINT"non-parametrical method. The" PRINT"weighting factor, u, is defined as:" PRINT:COLOR 15,1:PRINT"Wilkinson: u - 1":COLOR 7,0 PRINT:COLOR 15,1:PRINT"C.-Bowden: u - (Km + [S])/Vmax*[SJ":COLOR 7,0 . . PRINT:COLOR 15,1:PRINT"J. & Lumry: u - (Km + [S])“2/VM‘2*[S]“2":COLOR‘7,0 PRINT:PRINT"The non-parametrical method" PRINT"oalculates Km and Vmax estimates" PRINT"and uses the median value:" PRINT:COLOR 15,1:PRINT"Vmax(1.J) - [SJ - si]/[sj/vj - si/vi]" PRINT"Km(i,J) - si*[Vmax(i,j)/vi - 1]":COLOR 7,0 PRINT:PRINT PP$ ANS$=INKEY$:IF ANS$-"" THEN 6160 GOTO 5530 REM***********************************************§********X REM REM Option #E: End the program REM REM***1!-*************************************************XXX!- NIDTH 80:COLOR 7,0:CLS:END APPENDIX B This is a listing of the program LAGTIME, written in basic, which calculates the amount of coupling enzyme(s) needed to produce a desired lag time and the lag time for defined conditions. 100 200 300 N00 500 600 700 800 900 1000 1100 1200 1300 1N00 1500 1600 1700 1800 1900 2000 2100 2200 2300 2N00 2500 2600 2700 2800 2900 3000 3100 3200 3300 3N00 3500 REM Written by S. P. J. Brooks CLS: WIDTH NO: COLOR 15, N LOCATE 11, 7, 0 PRINT" " LOCATE ,7 PRINT" PRACTICALS OF COUPLING " LOCATE ,7 PRINT" " LOCATE ,7 PRINT" ENZYME THEORY " LOCATE ,7 PRINT" " LOCATE 20,1:PRINT"Copyright 1985, M.S.U." LOCATE 22,1:PRINT"Press any key to continue" ANYKEY$-INKEY$:IF ANYKEY$-"" THEN 1500 COLOR 7,0,0:CLS:WIDTH 80:LOCATE 1,1,0 PRINT:PRINT:PRINT:PRINT "This program calculates the parameters required to set up a" PRINT "successful coupled enzyme assay. Before one begins, a knowledge" PRINT"of the primary enzyme rate and the Km of the coupling enzyme(s)" PRINT"are necessary. You may then calculate the units of coupling" PRINT"enzyme(s) needed to obtain a predefined lag time or calculate " PRINT"the lag time when the units of coupling enzyme(s) are known." PRINT: PRINT"The equations are based on theory developed by S. P. J. Brooks," PRINT"T. Espinola and C. H. Suelter, Canadian Journal of Biochemistry and" PRINT"Cell Biology, 62, 9N5-955 and 956-963 (198N)." PRINT" --" PRINT"Four different assay systems can be analyzed: one and two coupling" PRINT"enzymes in which the first intermediate may or may not mutarotate." PRINT:PRINT:PRINT"Press any key to continue" ANYKEY$-INKEY$:IF ANYKEY$-"" THEN 3000 K1-3.8:K2-2.2:A-.N REM***§******§*************************************************** REM REM MENU REM 3600 3700 3800 3900 N000 N100 N200 N300 NNOO N500 N600 N700 N800 N900 5000 5100 5200 5300 5N00 5500 5600 5700 5800 5900 6000 6100 6200 6300 6N00 6500 6600 6700 6800 6900 7000 7100 7200 7300 7N00 7500 7600 7700 7800 7900 8000 8100 8200 8300 8N00 Appendix 8 (continued) REM****§************Niiiiiiif!*1!!!l1I1*********§********§*****{1*{111* WIDTH N0:CLS:LOCATE 1,1,0 PRINT:PRINT:PRINT:PRINT:PRINT"What do you want to do ?" PRINT:PRINT:PRINT"(A) Calculate lag times for given" PRINT" values of V2 and/or V3" PRINT:PRINT"(B) Calculate the amount of coupling" PRINT" enzyme to add to obtain a desired" PRINT" lag time" PRINT:PRINT"(C) Review the reaction schemes" PRINT" (alter some intrinsic parameters)" PRINT:PRINT"(D) Exit the program" ANYKEY$-INKEY$:IF (ANYKEY$<"A" OR ANYKEY$>"D") AND (ANYKEY$<"a" 0R ANYKEY$>"d") THEN N700 IF ANYKEY$="D" 0R ANYKEY$-"d" THEN N8N00 IF ANYKEY$-"A" 0R ANYKEY$-"a" THEN CAL=1 ELSE CAL=2 IF ANYKEY$-"C" 0R ANYKEY$-"c" THEN CAL =3 WIDTH N0:CLS:LOCATE 1,1,0 PRINT:PRINT:PRINT:PRINT:PRINT"Which system are you using ?" PRINT:PRINT:PRINT:PRINT:PRINT"(1) One coupling enzyme" PRINT:PRINT"(Z) Two coupling enzymes" PRINT:PRINT"(3) One coupling enzyme with" PRINT" mutarotation" PRINT:PRINT"(N) Two coupling enzymes with" PRINT" mutarotation" ANYKEY$'INKEY$:SYSaVAL(ANYKEY$):IF SYS<1 OR SYS>N THEN 5900 IF CAL-3 THEN 29300 ON SYS GOTO 6200,10000,1N500,17800 REMiflflflfifliflflifliliiiiiifii**************************************§** REM REM ONE ENZYME WITH NO MUTAROTATION REM REM*********i*******§******************************************** CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15.5 PRINT:PRINT"ONE ENZYME WITH NO MUTAROTATION:":COLOR 7,0 PRINT:INPUT"Enter the primary enzyme rate (mM/min) ";V1 PRINT:INPUT"Enter the KB value for the coupling enzyme'(mM) ";KB PRINT:INPUT"Enter the value of PB ";FB IF FB-1 THEN 7100 IF CAL - 2 THEN 7900 ' PRINT:INPUT"Enter the value of V2 (mM/min) ";V2 TFB - ‘KB/(VZ-V1)‘2*(FB*V1+V2*LOG(1-FB)) IF TFB<=0 THEN PRINT:GOSUB 28200: GOTO 8700 COLOR 15,1:PRINT PRINT "THE TIME REQUIRED TO REACH ";FB;" STEADY STATE IS: ";TFB;" MINUTES.":GOT0 8700 PRINT:INPUT"Enter the desired lag time (min) ";TFB SRB-LOG(1-FB)*KB/TFB-2*V1:SRCsV1“2+FB*KB*V1/TFB IF SRB“2*N*SRC<0 THEN PRINT:GOSUB 28200:GOTO 8700 SRU-SQR(SRB“2-N*SRC): ANS1-(-SRB-SRU)/2:ANSZ-(-SRB+SRU)/2 IF ANS1>V1 THEN V2-ANS1 IF ANSZ>V1 THEN V2-AN82 8500 8600 8700 8800 8900 9000 9100 9200 9300 9NOO 9500 9600 9700 9800 9900 10000 10100 10200 10300 10N00 10500 10600 10700 10800 10900 11000 11100 11200 11300 11N00 11500 11600 11700 11800 11900 12000 12100 12200 12300 12N00 12500 12600 12700 12800 12900 13000 Appendix B (continued) IF ANS1"D") AND (ANYKEY$<"a" OR ANYKEY$>"d") THEN 9100 IF ANYKEY$-"A" OR ANYKEY$-"a" THEN 3200 CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"0NE ENZYME WITH NO MUTAROTATION:":COLOR 7,0 PRINT:PRINT"Frimary enzyme rate - "V1"mM/min" IF ANYKEY$-"b" OR ANYKEY$-"B" THEN 7000 PRINT:PRINT"Coupling enzyme Km (value of K8) - ":KB;"mM" IF ANYKEY$-"c" OR ANYKEY$-"C" THEN 7100 PRINT:PRINT"FB - ":FB:GOTO 7300 REM§§¥§¥¥§*****§*!********************************************** REM REM TWO ENZYMES WITH NO MUTAROTATION REM REM********§********§*****************§*********§§************** CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT: PRINT"TWO ENZYMES WITH NO MUTAROTATION:": COLOR 7, O PRINT: INPUT"Enter the rate of the primary enzyme (mM/min) "1 :V1 PRINT: INPUT"Enter the values of KB, KC (mM) "1;KB, KC PRINT: INPUT"Enter the value of FC "1 ;FC IF FC-1 THEN 10900 IF CAL-1 THEN PRINT:INPUT"Enter the values of V2, V3 (mM/min) "1 :V2, V3: PRINT: OOTO 12700 PRINT: PRINT "Enter the desired value of t"; FC*100; "(min. ) "; INPUT TFCA TAU-TFCA/(1N9/9*FC-12.11) PRINT:INPUT"Enter the cost of enzyme II, IF P20 THEN 13200 COLOR 15,1 PRINT "THE TIME REQUIRED TO REACH ";FC;" STEADY STATE IS: III (cost/unit) ";P2,P3 ";T;" 13100 13200 13300 13N00 13500 13600 13700 13800 13900 1N000 1N100 1N200 1N300 1NN00 1N500 1N600 1N700 1N800 1N900 15000 15100 15200 15300 15N00 15500 15600 15700 15800 15900 16000 16100 16200 16300 16N00 16500 16600 16700 16800 16900 17000 17100 17200 17300 Appendix B (continued) MINUTES." PRINT"ERROR - +/- 10 I" COLOR 7,0:PRINT:PRINT"Enter A to return to menu" PRINT"Enter B to keep the primary enzyme rate" PRINT"Enter C to keep primary enzyme rate, KB, and KC" PRINT"Enter D to keep primary enzyme rate, KB, KC, and FC" ANYKEY$=INKEY$=IF (ANYKEY$<"A" OR ANYKEY$>"D") AND (ANYKEY$<"a" OR ANYKEY$>"d") THEN 13600 IF ANYKEY$-"A" OR ANYKEY$-"a" THEN 3200 CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"TWO ENZYMES WITH NO MUTAROTATION:":COLOR 7,0 PRINT:PRINT"Frimary enzyme rate - ";V1;"mM/min" IF ANYKEY$-"b" OR ANYKEY$-"B" THEN 10800 PRINT:PRINT"KB - ":KB;"mM","KC - ";KC;"mM" IF ANYKEY$-"C" OR ANYKEY$-"c" THEN 10900 PRINT:PRINT"FC - ";FC: GOTO 11100 REM!*§************§*****§****§****ii**************************** REM REM ONE ENZYME WITH MUTAROTATION 23:;XXNNXXNNNNNNNNNNHXNNNNNNNNNNINNXNXNNXNNNNXNNXXNNNNXXXIXXNNNX CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"0NE ENZYME WITH MUTAROTATION:":COLOR 7,0 PRINT:INPUT"Enter the primary enzyme rate (mM/min) ";V1 PRINT:INPUT"Enter the KB value for the coupling enzyme'(mM) ";KB PRINT:INPUT"Enter the value of EB ":FB IF FB-1 THEN 15NOO IF CAL -1 THEN'161OO ' PRINT:PRINT"Fnter the desired value of t";FB*100;"(min.) "; INPUT TFDESIRED ' ' TL-.01:TM=N99.99:TSTART-250:T-TSTART:GOSUB 22200:IF ERR1>0 THEN 16500‘ ' COLOR 15,1:PRINT:PRINT T;"mM/min of coupling enzyme is necessary to obtain the desired lag time":GOTO 16500 PRINT:INPUT"Enter the value of V2 (mM/min) ";V2 T-V2:COSUB 25700 COLOR 15,1:PRINT PRINT "THE TIME REQUIRED TO REACH ";FB;" STEADY STATE IS: ";TFBETA;" MINUTES." COLOR 7,0:PRINT:PRINT:PRINT"Enter A to return to menu" PRINT"Enter B to keep primary enzyme rate" PRINT"Enter C to keep primary enzyme rate, and KB of coupling enzyme" PRINT"Enter D to keep primary enzyme rate, KB of coupling enzyme, and FE" ANYKEY$=INKEY$zIF (ANYKEY$<"A" OR ANYKEY$>"D") AND (ANYKEY$<"a" OR ANYKEY$>"d") THEN 16900 IF ANYKEY$-"A" OR ANYKEY$-"a" THEN 3200 CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"0NE ENZYME WITH MUTAROTATION:":COLOR 7,0 PRINT:PRINT"Frimary enzyme rate - ";V1;"mM/min" 17N00 17500 17600 17700 17800 17900 18000 18100 18200 18300 18N00 18500 18600 18700 18800 18900 19000 19100 19200 19300 19N00 19500 19600 19700 19800 19900 20000 20100 20200 20300 20N00 20500 20600 20700 20800 20900 21000 21100 21200 21300 21N00 21500 21600 21700 21800 21900 22100 Appendix B (continued) IF ANYKEY$-"b" OR ANYKEY$-"B" THEN 15300 PRINT:PRINT"Coupling enzyme Km (value of KB) = ";KB;"mM" IF ANYKEY$-"c" OR ANYKEY$-"C" THEN 15NOO PRINT:PRINT"FB - ":FB:GOTO 15600 REM*******!**§***§§Ii**il§*§§*§ii§***********§**********§*****i* REM REM TWO ENZYMES WITH MUTAROTATION REM REMl-Nii**********************N*iii!l1!!-************************** CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT: PRINT"TWO ENZYMES WITH MUTAROTATION:1"1:COLOR 7, O PRINT: INPUT"Enter the rate of the primary enzyme (mM/min) "1 :V1 PRINT: INPUT"Enter the values of KB, KC (mM) "1,KB, KC PRINT: INPUT"Enter the value of FC "1 :FC IF FC<0 OR FC>-1 THEN 18700 IF CAL-1 THEN PRINT:INPUT"Enter the values of V2, V3 (mM/min) ";V2, V3: PRINT: GOTO 20NOO PRINT: PRINT "Enter the desired value of t"; FC*100; "(min. ) "; INPUT TFCA TAUsTFCA/(220/9*FC-19.62) PRINT:INPUT"Enter the'cost of enzyme II, III (cost/unit) ";P2,P3 IF P2O THEN 20900 COLOR 15,1 PRINT "THE TIME REQUIRED TO REACH ";FC;" STEADY STATE IS: ";T;" MINUTES." PRINT"ERROR - +/- 10 1" COLOR 7,0:PRINT:PRINT"Enter A to return to menu" PRINT"Enter B to keep the primary enzyme rate" PRINT"Enter C to keep primary enzyme rate, KB, and KC" PRINT"Enter D to keep primary enzyme rate, KB, KC, AND FC" ANYKEY$=INKEY$=IF (ANYKEY$<"A" OR ANYKEY$>"D") AND (ANYKEY$<"a" OR ANYKEY$>"d") THEN 21300 IF ANYKEY$-"A" OR ANYKEY$="a" THEN 3200 CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"TWO ENZYMES WITH MUTAROTATION:":COLOR 7,0 PRINT:PRINT"Frimary enzyme rate = "V1"mM/min" IF ANYKEY$-"b" OR ANYKEY$-"B" THEN 18600 PRINT:PRINT"KB - ";KB;"mM","KC = ";KC;"mM" 22000 IF ANYKEY$="C" OR ANYKEY$-"c" THEN 18700 PRINT:PRINT"FC - ";FC: GOTO 18900 22200 22300 22900 22500 22600 22700 22800 22900 23000 23100 23200 23300 23900 23500 23600 23700 23900 29000 29100 29200 29300 29900 29500 29600 29700 29800 29900 25000 25200 25300 25900 25500 25600 25700 25800 25900 26000 26100 26200 26300 26900 26500 26600 26700 26800 26900 27000 27100 27200 27300 27900 Appendix B (Oontinued) REM**§!*§*!***§****§******¥**§********************************** REM REM SUBROUTINE MINIMUM: FINDS THE VALUE OF T REM REM*****************!*§***************************************** Q-O:ERR1-0:ERR2-0 E-LOG(T/2)/LOG(2)+1 0N SYS GOSUB 29900, 29900, 25200, 26500: IF ERR2>0 THEN ERR1-1:RETURN T1- INT(T1*10000) IF T>TM OR T2 THEN PRINT. PRINT. GOSUB 28200. ERR1-1: RETURN IF T1-0 THEN RETURN IF T1>1E+1O AND E<.O1 THEN Q=Q+1.1:T=TSTART:OOTO 22800 IF Q<1 AND T1<0 THEN 23800 IF Q>1 AND T1>0 THEN 23800 T-O-T'23800 E-E-1zTaABS(2“E+T): GOTO 22900 REM************************************************************* REM REM EQUATION FOR TWO ENZYMES WITH NO MUTAROTATION REM REM************************************************************* c1-(V2-v1)‘2:02-(V3-v1)“2 IF(C1/V2*KB)<>(C2/V3*KC) THEN 2u800 V3-V3-V3/100 GOTO 2uu00 L-EXP(- T*C1/(V2*KB))- C1*KC*V3/(C2*KB*V2)*EXP(-T*C2/(V3*KC)) R-(1- C1*KC*V3/(C2*KB*V2))*(1-FC)*EXP(FC*V1/V3) T1-R- L 25100 RETURN REM************************************************************* REM REM EQUATION FOR ONE ENZYME WITH MUTAROTATION REM REM************************************************************* KBETA-KB/(1+K2/K1):V2=T M-(VZ-V1)/KBETA:BETASS-V1/M:SRB=K1+K2+M:SRC=9*K1*M SR-SRB“2-SRC:IF SR PRINT: COLOR COLOR COLOR v1 B ------------- > PRODUCT " COLOR ,0: PRINT" "; ,1:PRINT"ENZYME I"; ,O:PRINT" "; ,1:PRINT"ENZYME II":COLOR 0,7 PRINT:PRINT:PRINT"ASSUMPTIONS:":COLOR 15,0 PRINT:PRINT"(1) v1 is constant" PRINT"(Z) reverse reaction is negligible":COLOR 7,0 PRINT:PRINT:PRINT"Press any key to return to the menu" ANYKEY$=INKEY$:IF ANYKEY$-"" THEN 31500 GOTO 3700 CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"TWO ENZYMES WITH NO MUTAROTATION:":COLOR 15,0 PRINT:PRINT:PRINT"THE REACTION IS AS FOLLOWS: 1"1 :PRINT:PRINT PRINT" v2 * [8] V3 * [0]" PRINT" ---------------- PRINT" v1 KB + [B] KC + [C]":COLOR 15 u PRINT"S----*---> B -------------- > c -------------- > PRODUCT":COLOR,O PRINT" "; Appendix B (continued) 32500 COLOR,9:PRINT"/"; 32600 COLOR,0:PRINT" '3 32700 COLOR,9:PRINT"\" 32800 COLOR,0:PRINT" "; 32900 COLOR ,1:PRINT"ENZYME I"; 33000 COLOR ,O:PRINT" v; 33100 COLOR ,1:PRINT"ENZYME II"; 33200 COLOR ,O:PRINT" "3 33300 COLOR,9:PRINT"Q"; 33900 COLOR,0:PRINT" "; 33500 COLOR ,1:PRINT"ENZYME III"; 33600 COLOR,0:PRINT" "; 33700 COLOR, 9: 1PRINT"> R": COLOR 0, 7 33800 PRINT: PRINT: PRINT"ASSUMPTIONS:1"1:C0L0R 15, 0 33900 PRINT: PRINT"(I) v1 is constant" 39000 PRINT"(Z) reverse reactions are negligible" 39100 PRINT"(3) Fc >- .90":COLOR 7,0 39200 PRINT: PRINT"Press any key to return to the menu" 39300 ANYKEY$=INKEY$: IF ANYKEY$-"" THEN 39300 39900 GOTO 3700 39500 CLS: WIDTH 80:LOCATE 1,1,0:COLOR 15.5 39600 PRINT:PRINT"0NE ENZYME WITH MUTAROTATION:":COLOR 15,0 39700 PRINT:PRINT"THE REACTION IS AS FOLLOWS:":PRINT 39800 PRINT" a * V1":PRINT" "; 39900 COLOR 15,9:PRINT"S ‘ --> ALPHA" 35000 COLOR,0:PRINT" "; 35100 COLOR,9:PRINT"|"; 35200 COLOR,O:PRINT" n; 35300 COLOR,1:PRINT"ENZYME I"; 35900 COLOR,0:PRINT" n; 35500 COLOR,9:PRINT"|"; 35600 COLOR,0:PRINT" v; 35700 COLOR,9:PRINT"|\" 35800 COLOR,0:PRINT" n; 35900 COLOR,9:PRINT"|"; 36000 COLOR,0:PRINT" n, 36100 COLOR,9:PRINT"|"; 36200 COLOR,0:PRINT" v; 36300 COLOR,9:PRINT"|" 36900 COLOR,O:PRINT" n; 36500 COLOR,9:PRINT"|"; 36600 COLOR,0:PRINT" k1"; 36700 COLOR,0:PRINT" n; - 36800 COLOR,9:PRINT"|"; 36900 COLOR,0:PRINT" n; 37000 COLOR,9:PRINT"|"; 37100 COLOR,0:PRINT" k2 V2 + [BETAJn 37200 COLOR,0:PRINT" n; 37300 COLOR,9:PRINT"|"; 37900 COLOR,O:PRINT" n; 37500 COLOR,9:PRINT"|"; 37600 37700 37800 37900 38000 38100 38200 38300 38900 38500 38600 38700 38800 38900 39000 39100 39200 39300 39900 39500 39600 39700 39800 39900 90000 90100 90200 90300 90900 90500 90600 90700 90800 90900 91000 91100 91200 91300 91900 91500 91600 91700 91800 91900 Appendix B (continued) COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" " COLOR,0:PRINT" 'u COLOR,9:PRINT"|"; COLOR,0:PRINT" (I-a) * v1 "; COLOR,9:PRINT"\|"; COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" COLOR,0:PRINT" "; COLOR,9:PRINT"‘ PRODUCT" COLOR,0:PRINT" "; COLOR,9:PRINT"/"; COLOR,0:PRINT" "; COLOR,9:PRINT"\" COLOR,0:PRINT" "; COLOR,1:PRINT"ENZYME I"; COLOR,O:PRINT" "; COLOR,9:PRINT"Q"; COLOR,O:PRINT" "; COLOR,1:PRINT"ENZYME II"; COLOR,0:PRINT" "; COLOR,9:PRINT"> R":COLOR 0,7 LOCATE 18,1:PRINT"ASSUMPTIONS:":COLOR 15,0 PRINT:PRINT"(1) v1 is constant (2) reverse reaction is negligible" PRINT"(B) F BETA >- 0.9":COLOR 7.0 PRINT:PRINT"Fress any key to continue" ANYKEY$-INKEY$:IF ANYKEY$-"" THEN 90900 LOCATE 18,1:COLOR 0,7:PRINT”The following values are defined:"; COLOR 15,0:PRINT" " PRINT:PRINT"a - ":A;", k1 = ";K1;"(per min.), ";K2;"(per min. " ‘ PRINT" ’ " COLOR 7,0:PRINT"Press c to change a value" PRINT"Press m to return to the menu" ANYKEY$-INKEY$:IF ANYKEY$<>"O" AND ANYKEY$<>"C" AND ANYKEY$<>"M" AND ANYKEY$<>"m" THEN 91100 IF ANYKEY$-"M" 0R ANYKEY$-"m" THEN 3200 LOCATE 18,1:COLOR O,7:PRINT"Which value do you want to change:":COLOR 15,0 PRINT:PRINT"(A)‘a, fl PRINT:PRINT" " PRINT" " ANYKEY$-INKEY$:IF (ANYKEY$<"a" OR ANYKEY$>"O") AND (ANYKEY$<"A" OR ANYKEY$>"C") THEN 91700 LOCATE 20,1:PRINT" ' " PRINT"Enter the new value of "; KBETA + [BETA]" ‘> BETA '- k2 = (8) k1. (C) k2 92000 92100 92200 92300 92900 92500 92600 92700 92800 92900 93000 93100 93200 93300 93900 93500 93600 93700 93800 93900 99000 99100 99200 99300 99900 99500 99600 99700 99800 99900 95000 95100 95200 95300 95900 95500 95600 95700 95800 95900 96000 96100 96200 96300 96900 96500 96600 96700 96800 96900 97000 Appendix B (continued) IF ANYKEY$="a" 0R ANYKEY$="A" THEN INPUT"a ";A IF ANYKEY$-"B" 0R ANYKEY$-"b" THEN INPUT"k1 ";K1 IF ANYKEY$-"C" 0R ANYKEY$-"c" THEN INPUT"k2 ";K2 GOTO 90500 CLS:WIDTH 80:LOCATE 1,1,0:COLOR 15,5 PRINT:PRINT"THO ENZYMES WITH MUTAROTATION:":COLOR 15,0 PRINT:PRINT"THE REACTION IS AS FOLLOWS:":PRINT PRINT" v2 * [3]" PRINT” a * -------- " PRINT" v1 KB + [B]":PRINT" "; COLOR 15,9:PRINT"S ------ > B ---—---—---—----—--9—> ALPHA" COLOR,0:PRINT" 11; COLOR,1:PRINT"ENZYME I"; COLOR,0:PRINT" "; COLOR,9:PRINT"|": COLOR,0:PRINT" "; COLOR,1:PRINT"ENZYME II"; COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" 11; COLOR,9:PRINT"|\" COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" "; COLOR,9:PRINT"|" COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" v2 * [8] k1"; COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" "; COLOR,9:PRINT"|”: COLOR,0:PRINT" k2 V3 + [BETA]" COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" (1-a) * -------- "; COLOR,9:PRINT"|": COLOR,0:PRINT" "; COLOR,9:PRINT"|"; COLOR,0:PRINT" -------------- 11 COLOR,0:PRINT" n; COLOR,9:PRINT"|"; COLOR,0:PRINT" KB + [B] n; COLOR,9:PRINT"\|"; COLOR,0:PRINT" 9; COLOR,9:PRINT"|"; COLOR,0:PRINT" KBETA + [BETA]" COLOR,0:PRINT" "; COLOR,9:PRINT"‘ ------------------- > BETA -------------------- > 97100 97200 97300 97900 97500 97600 97700 97800 97900 98000 98100 98200 98300 98900 98500 98600 98700 98800 98900 Appendix B (continued) PRODUCT" COLOR,0:PRINT" "; COLOR,9:PRINT"/"; COLOR,0:PRINT" '% COLOR,9:PRINT"\" COLOR,0:PRINT" "; COLOR,1:PRINT"ENZYME II"; COLOR,0:PRINT" ‘ "; COLOR,9:PRINT"Q"; COLOR,0:PRINT" 'H COLOR,1:PRINT"ENZYME III"; COLOR,0:PRINT" "; COLOR,9:PRINT"> R" COLOR 0,7:GOTO 90000 REM*****************************************************&******* REM REM THIS IS THE END OF THE PROGRAM REM REM********************x**************************************** COLOR 7,0:WIDTH 80:END whi fun 10 20 30 90 50 60 70 8O 90 100 110 120 130 190 150 160 170 180 190 200 210 220 230 290 250 260 270 280 290 300 310 320 330 390 350 360 370 380 390 900 910 APPENDIX C This is the listing for NONLIN, a program, written in basic, ch allows one to calculate the parameters (up to 6) of a non-linear ction. REM NONLIN: Written by S. P. J. Brooks LG - 2. 302585093# ' PP$="press any key to continue" PPP$-"press 'c' to continue or 'p' to print" PPPP$="Is the printer turned on ?" PPPPP$-" " 55$.nxxxxxxxixxxxxx******xx*xxxxxxixxxxixxxxxxxxxxxxxxxxxx*xxxxxxxxxx ************fl CLS:LOCATE 1,1,0:WIDTH 90:COLOR 15,9:LOCATE 10,16:PRINT" " LOCATE 11, 16: PRINT" NONLIN "1:LOCATE 12,16:1PRINT" " LOCATE 19,1 :PRINT"copyright 1985, M. S. U. " LOCATE 21, 1, O: PRINT PP$ ANYKEY$= INKEY$: IF ANYKEY$="" THEN 120 DIM RESID(90),W(90),LPY$(75.95),A$(95),PT(90),X(91),Y(91),P(90,6). PPWP(6,6),PPW(6,90),YN(70).Q(6),B(6),VB(6),YHAT(91),BMAX(6),BMIN(6), BI(6) COLOR 7,0:CLS:LOCATE 1,1,0:PRINT:PRINT:PRINT"This program calculates the values of" PRINT:PRINT"parameters (max. of 6) for non-linear" PRINT:PRINT"cquations. This is accomplished using" PRINT:PRINT"the GauSS‘Newton algorithm adapted" PRINT:PRINT"from J. Fox in Linear Statistical" PRINT:PRINT"Models'and Related Models (1989)," PRINT:PRINT"John Wiley & Sons, N. Y." PRINT:PRINT:PRINT PP$ ' ' ENT$=INKEY$: IF ENT$="" THEN 220:NN=0:KK-0:FLAG6=0 REMxxxxxxxxxxxxixxxxxxxxxxxxxixxixxxxxxxxiixxx*xx********&* REM REM MENU REM REMxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxixxixxxxxxxxxxixxx CLS:LOCATE 1,1,0:COLOR 15,9:PRINT"You may do any of the following:" COLOR 7,0:PRINT:PRINT" ‘ 1. Input new data pairs" PRINT:PRINT" 2. Add data to existing data" PRINT:PRINT" 3. Edit and/or review the data" PRINT:PRINT" 9. Enter parameter estimates" PRINT:PRINT" 5. Calculate parameters," PRINT" plot data and residuals." PRINT:PRINT" H. Help menu/enter equation" PRINT:PRINT" E. Exit the program" PRINT:PRINT" F. Use data already on file" LOCATE 20,6,0:COLOR 15,2:PRINT"Did you remember to ":LOCATE 21,6,0:PRINT"enter yOur EQUATION ??":COLOR 7,0 ANS$=INKEY$= IF ANS$="" THEN 390 IF ANS$<>"f" AND ANS$<>"F" THEN 960 OPEN "NDATA" AS #1 LEN =8 920 930 990 950 960 970 980 990 500 510 520 530 590 550 560 570 580 590 600 610 620 630 690 650 660 670 680 690 700 710 720 730 790 750 760 770 780 790 800 810 820 830 890 Appendix C (continued) FIELD #1,9 AS 01$, 9 AS 02$:GET #1, 1:NN - CVI(Q1$) FOR 1-1 TO NN:RECN-I+1:CET #1, RECN:X(I)-CVS(Q1$):Y(I)-CVS(QZ$) NEXT I:CLOSE #1 LOCATE 18,9,0:COLOR 1S,3:PRINT"File data has been entered":COLOR 7,0 IF ANS$="e" 0R ANS$-"E" THEN 5560 IF ANS$-"h" 0R ANS$-"H" THEN 9980 A1- VAL(ANS$): IF A1<1 0R A1) 5 0R A1<>INT(A1) THEN 390 IF A1-9 AND NN<.5 THEN 520 IF A1<>5 THEN 580 IF A1-S AND NN>O AND KK>0 THEN S80 LOCATE 20,1,0:COLOR 15,9 IF NN=O THEN PRINT"You must enter data PRIOR ":PRINT"to selection of this option":GOTO 550 PRINT"You must enter parameter estimates":PRINT"PRIOR to selection of this option " COLOR 7,0:PRINT:PRINT PP$ AKEY$=INKEY$=IF AKEY$ - "" THEN 560 LOCATE 20,1,0:PRINT" ":PRINT" ":PRINT:PRINT" ":GOTO 380 ON A1 GOTO 590,590,1170,900,2090 REM§**************!*H***§****************§***************** REM REM option #1: Enter data pairs REM option #2: Add data to existing data REM REMi****§**i!******§§§**fl****&****§****i**§i!!!***********§ IF A1-2 THEN 730 IF NN<1 THEN 770 CLS:LOCATE 11,5:COLOR 15,9:PRINT"NOTE: This will erase all" LOCATE 12,5:PRINT"previously entered data 3!" PRINT:PRINT:PRINT"Press 'r' to return to the menu" PRINT"Press 'c' to continue" ANS$-INKEY$:IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"r" AND ANS$<>"R" THEN 710 COLOR 7,0:IF ANS$-"r" 0R ANS$-"R" THEN 230 ELSE 770 IF NN>-90 THEN CLS:COLOR 15,9:LOCATE 11,9:PRINT"9O data pairs is the limit !!":COLOR 7,0:PRINT:PRINT:GOTO 860 ES=NN+1:IF NN<12 THEN EN-13:GOT0 780 IF NN<27 THEN EN=27:GOTO 780 IF NN<=90 THEN EN-90:GOT0 780 EN=13:ES-1:LI-O IF A1-2 THEN LI-NN CLS:COLOR 15,1:PRINT"enter '*' after entering last data pair" LOCATE 3,1:PRINT" pair # ":LOCATE 3,19:PRINT" X ":LOCATE 3,26:PRINT" Y " ‘ FOR I-ES T0 EN:COLOR 15,1:LOCATE I+5-LI,1:PRINT I:COLOR 7,0:LOCATE I+5-LI,19:INPUT X$ IF X$-"*" THEN 860 ELSE X(I)-VAL(X$) LOCATE I+5-LI,26:INPUT Y(I):NEXT I IF I=19 THEN EN-27:ES-19:LI=13:GOTO 790 850 860 870 880 890 900 910 920 930 990 950 960 970 980 990 1000 1010 1020 1030 1090 1050 1060 1070 1080 1090 1100 1110 1120 1130 1190 1150 1160 1170 1180 1190 1200 1210 1220 1230 1290 Appendix C (continued) IF 1-28 THEN EN=90:ES=28:LI-27:GOTO 790 NN-I-1:COLOR 15,1:PRINT:PRINT NN"data pairs have been entered":GOSUB 5310 COLOR 7,0:PRINT:PRINT PP$ ANS$-INKEY$:IF ANS$-"" THEN 880 GOTO 230 REMifliiiiiiiiiiiifliiifiiiiiiifliiiiiii§****************§*I**§ REM REM OPTION #9: Enter parameter estimates REM REMHHHifliiiii*****§****¥*******§*§************************i IF KK<. 5 THEN FOR I-1 TO 6: 1BMIN(I)--1OOO: BMAX(I)-1000: NEXT I IF NN>6 THEN KMAX-6 ELSE KMAX-NN-1 CLS:LOCATE 3,1,0:PRINT"Enter the number of constants":PRINT"(parameters): maximum -";KMAX; INPUT KK IF KK<1 OR KK>KMAX THEN 970 COLOR 15,1:PRINT:PRINT"Enter the initial values for:":COLOR 7,0 PRINT:PRINT:FOR 1-1 TO KK:PRINT"B("I") "; INPUT BI(1):NEXT I' CLS:PRINT:PRINT"The maximum and minimum values for":PRINT"the constants have been set at: 1" COLOR 15,1.1LOCATE 6, 2: PRINT" # ":LOCATE 6, 15: PRINT" max "1 :LOCATE 6, 28: PRINT" min "1 :COLOR 7, 0 FOR 1-1 TO KK: LOCATE 7+1, 2: PRINT I:LOCATE 7+I,15:PRINT BMAX(I):LOCATE 7+I,28:PRINT BMIN(I):NEXT I LOCATE 7+I+2,1:PRINT"Are all values correct (y or n)":PRINT"If all values are incorrect enter 'a'" ENT$-INKEY$:1F ENT$<>"y" AND ENT$<>"Y" AND ENT$<>"n" AND ENT$<>"N" AND ENT$<>"a" AND ENT$<>"A" THEN 1070 ~- IF ENT$-"y" OR ENT$-"Y" THEN 230 ‘ IF ENT$<>"a" AND ENT$<>"A" THEN 1120 FOR 1-1 TO KK:LOCATE 7+1,1:PRINT"' II ' LOCATE 7+1, 2: COLOR 15, 9: 1PRINT I: LOCATE 7+I, 15: INPUT BMAX(I):LOCATE 7+1, 28: INPUT BMIN(I): COLOR 7, 0: NEXT 1: GOTO 1060 LOCATE 7+I+2, 11 .PRINT" " LOCATE 7+I+2, 1: INPUT"Which pair number is incorrect ";PN LOCATE 7+1+2,1:PRINT" ":IF PN<1 OR PN>KK THEN GOTO 1130 LOCATE 7+PN,1:PRINT" " " LOCATE 7+PN,2:COLOR 15,9:PRINT PN:LOCATE 7+PN,15:INPUT BMAX(PN):LOCATE 7+PN,28:INPUT BMIN(PN):COLOR 7,0:GOTO 1060 REMxxxNxxxxxxxxxxxxuxxxxxixxxxxxxxxxxu*x*x*x*****xxxxxxxxxx REM REM OPTION #3: Review the data REM OPTION #7: List residuals REM REM***********§****************i****!********************** IF NN<-13 THEN EN-NN:ES'1HLI-0 GOTO 1250 EN-13:ES=1:LI=O 1250 1260 1270 1280 1290 1300 1310 1320 1330 1390 1350 1360 1370 1380 1390 1900 1910 1920 1930 1990 1950 1960 1970 1980 1990 1500 1510 1520 1530 1590 1550 1560 1570 1580 1590 1600 1610 1620 1630 1690 1650 1660 1670 1680 1690 Appendix C (continued) CLS:LOCATE 1,1,0:COLOR 15,1:IF A1-3 THEN PRINT:PRINT" PAIR #":LOCATE 2,13:PRINT" X‘":LOCATE 2,26:PRINT" Y " IF A1-7 THEN LOCATE 2,1 :PRINT" # ":LOCATE 2, 8: 1PRINT" resid. "1 :LOCATE 2, 22: PRINT" 1 28. D. "1 :LOCATE 2, 32: PRINT" >28. D. ? " PRINT: FOR I-ES TO EN ' COLOR 15,1:LOCATE 3+I-LI,2:IF A1-3 THEN PRINT 1; COLOR 7,0:IF A1-3 THEN PRINT TAB(12) X(I) TAB(25) Y(I):GOTO 1390 PC-INT(RESID(I)/(SQR(SE)*2)*1000)/10 COLOR 15,1:LOCATE 3+I-LI,1:PRINT I: COLOR 7,0:PRINT TAB(6) RESID(I) TAB(23) PC:IF ABS(PC)<1OO THEN 1390 COLOR 15,9:LOCATE 3+1-LI,3S:PRINT" * ":COLOR 7,0 NEXT 1: IF A1-3 THEN 1550 REMNNNNNNNNNNNNNNiiiiiiiii*Niiiifi*******%**§******N******** REM REM continue listing the data REM REMNiiNiN-i*N!*************i****¥*************************** IF NNsI-1 THEN 1960 LOCATE 9+I-LI:PRINT PP$ ANS$-1NKEY$:IF ANS$-"" THEN 1920 IF NN<-27 AND I<28 THEN EN=NN: ES=13: LI=12: GOTO 1250 IF NN>27 AND I<28 THEN EN-27: ES-13: L1-12: GOTO 1250 EN=NN: E3827: LI= 261 GOTO 1250 REMNNxxxxxxxxxxxxxxxxxxxxxrxxxxxixxxxxx******x****x****x*** REM REM stop listing the data REM REM:xxxxxxxxxxuxxxxxxxxxxuxxxiixxxfianxuxxxxxxrxxxxxxxuxxxxx IF NN+1-1 THEN COLOR 15,1:PRINT:PRINT:PRINT"sequence - ";BB$:PRINT AA$: COLOR 7, 0 PRINT. PRINT PPP$ ANS$-INKEY$:1F ANS$<>"c" AND ANS$<>"C" AND ANS$<>"p" AND ANS$<>"P" THEN 1530 IF ANS$-"p" OR ANS$-"P" THEN 1850 ELSE COLOR 7,0:GOTO 3500 REM**§*****iiiiiiiifiiifliiifiiififliiiifliiiiiiiiiii§****§**§*** REM REM continue with option #2 REM REMNNNN11!********************************N****************** LOCATE 5+I-LI,1:PRINT"Press 'e' to edit data" PRINT"Press 'd' to delete data" PRINT"Press 'c' to continue" ANS$=INKEY$: IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"e" AND ANS$<>"E" AND ANS$<>"d" AND ANS$<> "D" THEN 1630 IF ANS$<>"c" AND ANS$<> "C" THEN 1690 ' IF 1aNN+1 THEN 230 IF 1-19 AND NN<27 THEN EN -NN: ES-13: LI-12 GOTO 1250 IF I-19 AND NN>27 THEN EN -27:ES-13:L1-12:GOT0 1250 EN=NN: ES - 271 LI=26= GOTO 1250 LOCATE I+5’LI:COLOR 15,9:PRINT"Which data pair do you want to" Appendix C (continued) 1700 COLOR 7.0 1710 PRINT" " 1720 PRINT" " 1730 LOCATE 6+I-LI,1:COLOR 15.9 1790 IF ANS$(>"e" AND ANS$<>"E" THEN 1820 1750 INPUT"adit ";N2 1760 IF N2EN THEN 1730 1770 COLOR 7,0:LOCATE 3+N2-LI,1:PRINT" " 1780 COLOR 15,9:LOCATE 3+N2-LI,2:PRINT N2 1790 LOCATE 3+N2-LI,12:INPUT" ";X(N2) 1800 LOCATE 3+N2-LI,25:INPUT" ";Y(N2) 1810 COLOR 7,0:GOSUB 5310: GOTO 1170 1820 INPUT"delete ";N2:COLOR 7,0 ' 1830 IF N2<1 0R N2>NN THEN 1730 1890 NN-NN-1zFOR I-N2 T0 NN:Y(I)-Y(I+1):X(I)=X(I+1):NEXT I:GOSUB ' 5310:00T0 1170 1850 REM*******§************iii}********N************************** 1860 REM 1870 REM print results to line printer 1880 REM 1890 REM*********************************************************** 1900 LOCF-8+I-LI 1910 LOCATE LOCF:PRINT PPPPP$:COLOR 15,9:LOCATE LOCEzPRINT PPPP$:COLOR . 7’0 1920 PRINT:PRINT PP$ 1930 ANS$-INKEY$:IF ANS$=""THEN 1930 1990 LPRINT:LPRINT EE$ 1950 LPRINT:LPRINT:LPRINT"Results of NONLIN calculations: ";CC$ 1960 LPRINT:LPRINT:LPRINT"pair #" TAB(17) "X value" TAB(39) "Y value"~ ‘ TAB(51) "residual" TAB(68) "% of 2 S. D." 1970 LPRINT" ------ " TAB(17) "-----9-" TAB(39) " ------- " TAB(51) " ------ ‘ --" TAB(68) " ----------- " 1980 LPRINT: FOR I=1 TO NN 1990 PC-INT(RESID(I)/(SQR(SE)*2)*1000)/10 2000 LPRINT I TAB(17) X(I) TAB(39) Y(I) TAB(51) RESID(I) TAB(68) PC 2010 NEXT 1 2020 LPRINT:LPRINT:LPRINT"Sequence of residuals a ";BB$:LPRINT AA$ 2030 LPRINT:LPRINT EE$=GOT0 3500 20140 REM§********************************************************** 2050 REM 2060 REM Option #5: Parameter estimation --> leads to sub-menu 2070 REM 2080 REM************!***************************************§****§* 2090 CLS:LOCATE 3,1,0:PRINT"Nhat type of weighting ?" 2100 COLOR 7,0:LOCATE 6,1:PRINT"(a) constant absolute error" 2110 PRINT" (constant standard deviation)":PRINT" "; 2120 COLOR 15,1:PRINT"w - 1":COLOR 7,0 2130 LOCATE 10,1:PRINT"(b) constant relative error" 2190 PRINT" ‘ (proportional standard deviation)":PRINT" "; 2150 COLOR 15,1:PRINT"H - (1/y)“2":COLOR 7.0 2160 LOCATE 15,1:PRINT"(c) Intermediate case":PRINT" "; 2170 2180 2190 2200 2210 2220 2230 2290 2250 2260 2270 2280 2290 2300 2310 2320 2330 2390 2350 2360 2370 2380 2390 2900 2910 2920 2930 2990 2950 2960 2970 2980 2990 2500 2510 2520 2530 2590 2550 2560 2570 2580 2590 2600 2610 Appendix C (continued) COLOR 15,1:PRINT"w - 1/y": COLOR 7, 0 06$=INKEY$: IF (06$("a" 0R 06$)"c") AND (06$("A" OR 06$)"C") THEN 2180 FOR I- 1 T0 NN IF 06$-"a" OR 06$-"A" THEN N(I)-1:CC$-"constant absolute error" IF 06$-"b" OR O6$-"B" THEN N(I)- (1/Y(I)) 2: CC$-"constant relative error" IF O6$="c" OR O6$-"C" THEN W(I)-1/Y(I): CC$-"inbetween case" NEXT I. FOR 191 T0 KK: B(I)=BI(I): NEXT I: GOSUB 5960: SS-O: FOR I-1 TO NN SS=SS+W(I)*(Y(I)-YHAT(I))“2:NEXT I:C=100:SSO-SS:ITER=0:ER$-"":CLS ITERtITER+1:IF ITER>-20 THEN ER$="Max1mum of 20 iterations !!":GOTO 3120 LOCATE 10, 9: PRINT"iteratlon #"; ITER: LOCATE 13, 9: 1PRINT"SS -"1HSS LOCATE 19, 9: 1PRINT"# to zero -";C: LOCATE 16, 9: 1PRINT"Press 'q' to quit" ‘ REM******************************§************ REM REM Calculate a P(NN*KK) partial derivative matrix REM REM**xxxxxxx**xxxxxxxxxxxxxxxxxxxxxxxxxxxxixxx FOR J-1 T0 KK: B(J)-1.02*B(J):GOSUB 5960 FOR 1-1 TO NN:YN(I)-YHAT(I):NEXT I B(J)=B(J)*.98/1.02:GOSUB 5960 B(J)=B(J)/‘98 " FOR 181 T0 NN: P(I, J)= (YN(I)- YHAT(I))/(. 09*B(J)): NEXT I NEXT J REMxxxxx*******x*xxxxxxxuxxxx*******x*n***x*** REM REM Calculate the values of Q (matrix corrections) REM REM (1) Calculate the P'WW'P matrix labelled as PPWP(KK*KK) REM Rngxxxxxxxxxxuxxxxxxxxxxxxxixxixxxxxxxxxxxxxx FOR I-1 T0 KK:FOR J-1 T0 NN PPW(I,J)=P(J,I)*W(J):NEXT J:NEXT I FOR I-1 TO KK:FOR J-1 T0 KK:PPWP(I,J)=0:FOR K-1 TO NN PPWP(I,J)-PPWP(I,J)+PPW(I,K)*P(K,J):NEXT K:NEXT J:NEXT I REMi****1X11X1l1*********************************** REM REM (2) Invert the PPWP matrix REM REM*****xxxxxixuxxxxxxxxxxxxxxxx*xxx**x****x** IF KK-1 THEN PPWP(1, 1)-1/PPWP(1, 1): GOTO 2890 FOR L-1 T0 KK: DD-O ‘ FOR K-1 T0 KK:DD-DD+PPWP(L,K)*PPWP(L,K):NEXT K DD=SQR(DD):NEXT L FOR L-1 T0 KK:YN(L+20)-L:NEXT L L80 L-L+1:IF L>KK THEN 2780 CC=0:M=L 2620 2630 2690 2650 2660 2670 2680 2690 2700 2710 2720 2730 2790 2750 2760 2770 2780 2790 2800 2810 2820 2830 2890 2850 2860 2870 2880 2890 2900 2910 2920 2930 2990 2950 2960 2970 2980 2990 3000 3010 3020 3030 3090 3050 3060 3070 3080 3090 Appendix C (continued) FOR K-L T0 KK: IF ABS(CC)>-ABS(PPHP(L,K)) THEN 2690 M-K:CC-PPWP(L,K) NEXT K IF L-M THEN 2700 K-YN(M+20):YN(M+20)-YN(L+20):¥N(L+20)=K FOR K-1 T0 KK:S=PPHP(K,L) PPHP(K;L)-PPWP(K,M):PPHP(K,M)-S NEXT K PPHP(L,L)-1 FOR M-1 T0 KK:PPHP(L,M)-PPWP(L,M)/CC:NEXT M FOR M-1 T0 KK:IF L-M THEN 2760 CC-PPWP(M,L):IF CC-O THEN 2760 PPWP(M,L)-0 FOR K-1 T0 KK:PPHP(M,K)-PPWP(M,K)-CC*PPWP(L,K):NEXT K NEXT M" GOTO 2600 L1-0 L1-L1+1:IF L1>KK THEN 2890 IF YN(L1+20)-L1 THEN 2790 M-L1 M-M+1 IF YN(M+20)-L1 THEN 2850 IF KK>M THEN 2820 YN(M+20)-YN(L1+20) FOR K-1 T0 KK: CC-PPWP(L1,K) PPWP(L1,K)-PPWP(M,K):PPWP(M,K)-CC:NEXT K YN(L1+20)-L1:GOT0 2790 REM****i*i§*******i**{X*X!§*********§**§*fli**§ REM REM (3) Calculate the P'HW'E matrix labelled as YN(KK) REM REMxxnxxxxxuxxxxxxxixxxnxuxxxunxxxaxnnuanuixaa GOSUB 5960 FOR I-1 T0 KK:YN(I)-0:FOR K-1 T0 NN YN(I)-YN(I)+PPW(I,K)*(Y(K)-YHAT(K)):NEXT K:NEXT I REM************************Iiiifififliiiiiiififlifli REM REM calculate the parameters and check the results REM REM******************************§Hi*§***§**§* ML-1:FOR I - 1 T0 KK:Q(I)-0:YN(I+KK)-B(I):FOR K-1 T0 KK Q(I)-Q(I)+PPWP(I,K)*YN(K):NEXT K:NEXT I:ITERA-O C=O:FOR I-1 T0 KK:C-ABS(Q(I)/B(I))+C:B(I)-YN(I+KK)+ML*Q(I):IF B(I)BMAX(I) THEN 3090 NEXT I:GOSUB 5960:SS-O:FOR I-1 T0 NN:SS-SS+W(I)*(Y(I)- YHAT(I))‘2:NEXT I ' QU$-INKEY$:IF QU$<>"q" AND QU$<>"Q" THEN 3080 FOR I-1 T0 KK:B(I)-YN(I+KK):NEXT I:GOTO 3120 IF SS=20 THEN ER$-"Local minimum found !!":GOTO 3070 3100 3110 3120 3130 3190 3150 3160 3170 3180 3190 3200 3210 3220 3230 3290 3250 3260 3270 3280 3290 3300 3310 3320 3330 3390 3350 3360 3370 3380 3390 3900 3910 3920 3930 3990 3950 3960 3970 3980 3990 3500 3510 Appendix C (continued) ML-ML/2:OOTO 3090 SSO=SS=IF c> .00001 THEN 2250 REM!!!%*******§**§!******%*§§*****XH{fluX1111!!!» REM REM Run has converged 1!! REM REMit-iiiiiiii***********§******111111111!XHXN'X-i SE=SSO/(NN-KK):FOR 1-1 T0 KK:VB(I)-SQR(ABS(SE*PPHP(I,I))):NEXT I CLS:PRINT:PRINT:COLOR’15,9:PRINT"Results:"; COLOR 7,0:PRINT " "+CC$:PRINT:COLOR 15,9:PRINT ER$:COLOR 7,0:PRINT ' PRINT:FOR 1:1 To KK:PRINT "B(";I;") -";B(I);" +/-";VB(I):NEXT I PRINT:PRINT:PRINT PPP$ ANS$-INKEY$:IF ANS$<>"c" AND ANS$<>"C" AND ANS$<>"p" AND ANS$<>"P" THEN 3220 IF ANS$:"c" OR ANS$-"C" THEN 3500 REM!ii*Xliriifiiiiiiiifiiiii******!*****INix-«HEN****************** REM REM print to line printer REM REM**************§********§*************Xii-iXX-XXXHX-XXX'XXX-XXX!” LOCATE 9+I,1:PRINT PPPPP$:COLOR 15,9:LOCATE 9+I,1:PRINT PPPP$:COLOR‘7.0 1 PRINT PP$ ANS$-INKEY$:IF ANS$-"" THEN 3310 LPRINT:LPRINT EE$ LPRINT:LPRINT"Results of NONLIN calculations: ";CC$ LPRINT:LPRINT:LPRINT TAB(1) "pair #" TAB(20) "X value" TAB(90) "Y value" TAB(60) "Yhat" ‘ LPRINT TAB(1) " ------ " TAB(20) " ------- " TAB(90) " ------- " TAB(OO) 11--...." 1 LPRINTzFOR I-1 T0 NN:LPRINT TAB(B) I TAB(20) X(I) TAB(90) Y(I) TAB(60) YHAT(I):NEXT I LPRINT:LPRINT:LPRINT "# iterations - ";ITER,"SS - ";SS:LPRINT LPRINT:FOR I-1 T0 KK:LPRINT "B(";I;") -";B(I);" +/-";VB(I):NEXT I LPRINT:LPRINT"The covariance matrix is as follows:" LPRINT:LPRINT" var(1,1), var(1,2), var(1,3), LPRINT:FOR I=1 T0 KK:LPRINT"var("I",1)"; ' IF KK-1 THEN LPRINT" ";PPWP(I,1) IF KK=2 THEN LPRINT" ";PPHP(I,1);" ";PPWP(I,2) IF KK-3 THEN LPRINT" ";PPWP(I,1);" ";PPWP(I,2);" ";PPWP(I,3) IF KK=9 THEN LPRINT" ";PPWP(I,1);" ";PPWP(I,2);" ";PPWP(I,3);" ";PPWP(I,9) ‘ IF KK-S THEN LPRINT" ";PPWP(I,1);" ";PPWP(I,2);" ";PPWP(I,3);" ";PPWP(I,9);" ";PPWP(I,5) IF KK=6 THEN LPRINT" ";PPWP(I,1);" ";PPWP(I,2);" ";PPWP(I,3);" ";PPWP(I,9);" ";PPWP(I,5);" ";PPWP(I,6) NEXT I LPRINT:LPRINT EE$ REM*****x***x******x*****x******xxxxxxxx**x*******xxxx******xx REM 3520 3530 3590 3550 3560 3570 3580 3590 3600 3610 3620 3630 3690 3650 3660 3670 3680 3690 3700 3710 3720 3730 3790 3750 3760 3770 3780 3790 3800 3820 3830 3890 3850 3860 3870 3880 3890 3900 3910 3920 3930 3990 Appendix C (continued) REM Sub-menu REM REMxxxxxxxxxxixiixxxxxxxixxxxxxxxixxxxxxxxxxxxxxxxxxxxxxxxxxxx CLS: LOCATE 2, 1, 0: COLOR 15, 9: 1PRINT"You may do any of the following: COLOR 7, 0: PRINT:PRINT" PRINT:PRINT" PRINT:PRINT" 5. Recalculate parameters" 6. Plot the data" 7. List residuals" PRINT:PRINT" 8. Plot residuals" PRINT:PRINT" R. Return to main menu" ANS$-INKEY$: IF ANS$="" THEN 3610 IF ANS$-"r" OR ANS$-"R" THEN 230 A1- VAL(ANS$): IF A1<5 0R A1) 8 OR A1<>INT(A1) THEN 3610 A11-A1- 9: 1ON A11 GOTO 2090, 3650, 9130, 9130 REMNxxxxxxxxxxixxxixxxxixxxxxxxxixxxxH»xxxxxxxixxixxxxxxxrxxx* REM REM OPTION #6: Plot the data REM REMxxxxixxixxxaxxxxxxxxxxxxxxxxxxxxxxxxxxxxixxxxxxxxxixxxxxxxx CLS: LOCATE 9, 1,0: COLOR 15,1:PRINT"How do you want to plot the data 7": COLOR 7, 0 ' LOCATE 7,1 :PRINT"(a) Y versus X" PRINT:PRINT"(b) 1/Y versus 1/X" PRINT:PRINT"(C) Y versus log(x)" PRINT:PRINT"(d) log(Y) versus log(X)" O6$-INKEY$:IF (06$("a" OR O6$>"d") AND (O6$<"A" OR O6$>"D") THEN 3750 TEMP - 1:TEMP1-1:FOR I-2 TO NN IF O6$-"b" OR O6$-"B" THEN IF 1/Y(I)>1/Y(TEMP) THEN TEMP=I IF O6$<>"b" AND O6$<>"B" THEN IF Y(I)>Y(TEMP) THEN TEMP =II IF O6$-"d" OR O6$-"D" THEN IF Y(I)I THEN TEMP-PT(I): PT(I)=PT(I1): PT(I1)=TEMP 9290 NEXT I 9250 M9-O:N9=O:U9=0:A2-O:RB$-"" 9260 FOR I=1 T0 NN: IF RESID(PT(I))0 THEN N9-N9+1: A3= 2: BB$=BB$+"+" 9280 IF A3<>A2 THEN U9-U9+1 9290 A2-A3: NEXT I 9300 IF M9+N9<8 THEN AA$-"Too few points to analyse randomness":00T0 9970 9310 IF N9INT(K9) THEN K9-(I+1)/2:GOTO 9380 9350 9360 9370 9380 9390 9900 9910 9920 9930 9990 9950 9960 9970 9980 9990 9500 9510 9520 9530 9590 9550 9560 9570 9580 9590 9600 9610 9620 9630 9690 9650 9660 9670 9680 9690 9700 9710 9720 9730 9790 9750 9760 9770 9780 9790 9800 9810 9820 Appendix C (continued) A3-M9-1:A9-K9-1:GOSUB 5180:C1-C9 A3-N9'1:A9=K9-1:OOSUB 5180:C2-C9 FU - 2*C1*C2:GOT0 9930 A3-M9-1:A9-K9-1:GOSUB 5180:C1-C9 A3-N9-1:A9=K9P2:GOSUB 5180:C2-C9 A3-M9- 1 :A9-K9- 2: GOSUB 5180: C3-C9 A3-N9- 1 :A9-K9- 1: GOSUB 5180: C9-C9 FU=C1*C2+CB*C9 FT-FT+FU: NEXT I A3-N9+M9:A9=M9:GOSUB 5180: FT-FT/C9:Z9-1 IF FT>. 05 THEN AA$-"The sequence is random (p>0. 95)" IF FT<-. 05 THEN AA$="The sequence is non random (p>0. 95)" IF A1t7 THEN 1170 REMXH********l**************************§***§*I********§*1X1 REM REM Option #8: Plot residuals (continued) REM REM***********§*****XXX*****************%**************** B1-Y(1): 82= ABS(RESID(1)) FOR la 2 TO NN IF ABS(RESID(I))>BZ THEN BZ=ABS(RESID(I)) IF Y(I)>B1 THEN B1=Y(I) NEXT I:SCREEN 1,0 COLOR 0,1:PRINT"Sequence . ";BB$:PRINT AA$ FOR I-1 T0 8:PRINT:NEXT I:PRINT"R":PRINT"e":PRINT"s":PRINT"i"; PRINT TAB(28)" Y ":PRINT"d" LOCATE 23,1:PRINT PPP$ LINE (33. 30)- (30, 175), 2, BF LINE (30,109) (300, 101), 2, BF FOR 1-1 T0 NN XCORD - 33 + Y(PT(I))/B1*267 YCORD 3102 - (RESID(PT(I))/82)*72: CIRCLE (XCORD,YCORD),2,1,,,1 NEXT I ANS$= INKEY$:IF ANS$<>"C" AND ANS$<>"C" AND ANS$<>"p" AND ANS$<>"P" THEN 9680 IF ANS$-"c" 0R ANS$="C" THEN SCREEN 0, 1: COLOR 7, 0: GOTO 3500 LOCATE 23,1 :PRINT PPPPP$: LOCATE 22,1 :PRINT PPPP$: LOCATE 23,1 :PRINT PP$ ANS$- INKEY$:IF ANS$="" THEN 9710 REM*********************************************************** REM REM plot residuals on line printer REM REM!!-***************§§****X1111!!!****************il************ SCREEN 0,1:COLOR 7,0:CLS:FOR I=0 T0 90: A$(I)-" ":NEXT I A$(18)=" r ":A$(17)-" e ":A$(16)-" s ":A$(15)-" i ":A$(19)=" d ":A$(13)-" u ":A$(12)=" a ":A$(11)=" l " FOR X-O T0 70: FOR Y=0 T0 30: LPY$(X, Y)-" "1 :NEXT Y: NEXT X FOR X-1 T0 70: IF INT(X/10)-X/10 THEN A1$-"|" ELSE A1$-" " LPY$(X, 15)-A1$: NEXT X FOR Y=0 T0 30: IF INT(Y/3)=Y/3 THEN A1$="~" ELSE A1$="|" 9830 9890 9850 9860 9870 9880 9890 9900 9910 9920 9930 9990 9950 9960 9980 9990 5000 5010 5020 5030 5090 5050 5060 5070 5080 5090 5100 5110 5120 5130 5190 5150 5160 5170 5180 5190 5200 5210 5220 5230 5290 5250 5260 5270 Appendix C (continued) LPY$(O,Y)-A1$:NEXT Y FOR I=1 TO NN Y=(RESID(PT(I))+82)/BZ*19. 5: X=Y(PT(I))/B1*7O LPY$(INT(X+. 5), INT(Y+. 5))-"0" NEXT I ‘ LPRINT: LPRINT: LPRINT EE$: LPRINT: LPRINT"Craph of residuals versus velocityz": LPRINT" ----------------------------------- "1 HLPRINT LPRINT FOR Y=30 TO 0 STEP -1 LP5$-"":FOR x=0 TO 70:LP5$-LP5$+LPY$(X.Y):NEXT x LPRINT " "+A$(Y)+LP5$:NEXT Y LPRINT: LPRINT" Y value" LPRINT: LPRINT"x interval - "1HB1/7 LPRINT"y interval . "1 ;B2/15*3 LPRINT: LPRINT"Residuals calculated using: "1 :CC$ LPRINT: LPRINT"sequence of residuals - "1 HBB$ LPRINT AA$ LPRINT:LPRINT EE$:GOTO 3500 9970 SCREEN 0,1:COLOR 7,0: GOTO 3500 REM*******§****iiiifiiiifiiifliifiiiiiiifiiiiiii!*§***§************ REM REM Option #H: Help menu REM REM§*************fl*************************************i****** CLS:LOCATE 2,1,0:PRINT"Equations are entered starting at line" PRINT" in the following format:" PRINT:GOLOR 15,1:PRINT"yhat( ) - f{B(1), 3(2), ..., x( )} COLOR 7, O: PRINT:PRINT"An example is as follows:"" PRINT: COLOR 15,1:PRINT" yhat( ) - B(1)+exp(B(2)*x( ))":COLOR 7, 0 ' PRINT:PRINT"Note that the variable is always - " COLOR 15,9:LOCATE 5,6:PRINT "JJ":LOCATE 5,33:PRINT "jj":LOCATE 9,11:PRINT "JJ":LOCATE 9,33:PRINT "JJ":LOCATE 11,36:PRINT"JJ":COLOR 7,0 PRINT"and that the equation has a line number." PRINT"To enter more than one equation, number" PRINT"the lines in increments of 1. For" PRINT"exampIe: , . , ..." PRINT: PRINT PP$ ‘ COLOR 15, 2: LOCATE 3,1 :PRINT "5520": LOCATE 9,1 :PRINT "5520": LOCATE 16, 10: PRINT "5520": LOCATE 16, 16: PRINT"5521": LOCATE 16, 22: PRINT"5522": COLOR 7. O ' ANS$=INKEY$: IF ANS$-"" THEN 5160 GOTO 230 REMXHXHX11**************************{1*******X**1!!!*************** REM REM Factorial subroutine REM REM***ilifiiiiiii*************************§******************** J1-1:J2=1:J3=1:AS-(A3-A9) IF A3<-O THEN A3-1 IF A9<-O THEN A9-1 IF A5<=O THEN A5-1 FOR I3- A3 TO 1 STEP -1:J1=J1*I3:NEXT I3 5280 5290 5300 5310 5320 5330 S3NO 5350 5360 5370 5380 5390 suoo 5u1o suzo 5u3o suuo suso suso suvo suao 5u9o 5500 5510 5520 5530 ssuo 5550 5560 5570 5580 5590 5600 5610 Appendix C (continued) FOR I3 -Afl T0 1 STEP -1:J2-J2*I3:NEXT I3 FOR I3=A5 T0 I'STEP -1:J3-J3*I3:NEXT I3 C9-J1/(J2*J3):RETURN REM§§*x*********x**x**********x********x***********x**xx*** REM REM Sorting subroutine REM REM§§¥§¥§§§§§§§§*********************§********************N FOR I3=1 TO NN-1: Il- -I3: LE-X(13) FOR Iu-I3+1 T0 NN: IF X(Ih)